Cylinder absolute games on solenoids

Let $A$ be any affine surjective endomorphism of a solenoid $\Sigma_{\mathcal{P}}$ over the circle $S^1$ which is not an infinite-order translation of $\Sigma_{\mathcal{P}}$. We prove the existence of a cylinder absolute winning (CAW) subset $F \subset \Sigma_{\mathcal{P}}$ with the property that for any $x \in F$, the orbit closure $\overline{\{ A^{\ell} x \mid \ell \in \mathbb{N} \}}$ does not contain any periodic orbits. Dimension maximality and incompressibility of CAW sets is also discussed for a number of possibilities in addition to their winning nature for the games known before.


Introduction
Let P be an (finite or infinite) ordered set of prime numbers with p 1 < p 2 < · · · and X P,n be the restricted product space where ′ denotes that for each element x ∈ X P,n , the entries x p ∈ Z n p for all but finitely many p's. A P-solenoid of topological dimension n is the quotient space Σ P (n) := XP,n /∆(R n ), where R is the ring Z[{p −1 | p ∈ P}] whose n-fold product is embedded diagonally in X P,n as a uniform lattice. We call the quotient map X P,n → Σ P (n) to be Π. Solenoids are compact, connected metrizable abelian groups. They have sometimes been called "fractal versions of tori" [22]. When P is a finite set of cardinality l − 1, the Hausdorff dimension of X P,n (and therefore of Σ P (n) too) under the natural metric given by (2.1) is nl. This also implies that the dimension is infinite when P is so, as the increasing sequence of finite product spaces associated with the finite truncations of P are isometrically embedded inside X P,n .
The set of endomorphisms of Σ P := Σ P (1) is precisely the ring R whose elements act multiplicatively componentwise. An affine transformation A : Σ P → Σ P is meant to denote the map x → (m/n)x + a (1.2) where m/n ∈ R \ {0} and a ∈ Σ P . It is well known that when A = m/n is a surjective endomorphism of the solenoid, it acts ergodically on Σ P iff m/n / ∈ {0, ±1} [24,Proposition 1.4]. We also learn from Berend [4,Theorem 3.2] that for every compact group G, any semigroup of its affine transformations lying above an ergodic semigroup of surjective endomorphisms is ergodic as well. This gives us a sufficient condition for the transformation Ax = (m/n)x + a to be ergodic, namely that m/n = ±1. Ergodicity of the action guarantees that almost all orbits of A are dense in Σ P . However, just like Dani [8], this work is concerned with understanding the complementary set. Given an affine transformation A, we will like to know the set of points of Σ P whose A-orbits remain away from periodic B-orbits for all B ∈ R \ {±1}.
When P is the set consisting of all the primes in N, the space Σ P is called the full solenoid over S 1 with the field Q being its ring of endomorphisms. Let B be a non-zero rational number. The growth of the number of B-periodic orbits as a function of the period is determined by the entropy of the action on Σ P . Latter has been computed by Juzvinskiȋ [13] first and recovered by Lind and Ward in [18] who explained it to be the sum of the Euclidean and the p-adic contributions. In fact, they achieve it for all automorphisms of solenoids over higher-dimensional tori as well. We remark that each such epimorphism lifts uniquely to a homomorphism from X P := X P,1 to itself, which we shall continue to denote by the same rational number. For an affine transformation, we however have a choice involved in terms of a representative for the translation part a.
Let y ∈ Σ P be arbitrary and A be a (surjective) affine transformation of Σ P as in (1.2) with either m/n = 1 or a = 0. We intend to show that the set of points x ∈ X P whose forward orbit under the map x → Ax maintains some positive distance from the 1-uniformly discrete subset Π −1 ({y}) ⊂ X P is cylinder absolute winning (CAW) in a similar sense as Fishman, Simmons, and Urbański [12]. Once we have this, we can take intersection of countably many of these sets to conclude about A-orbits which avoid neighbourhoods of all periodic orbits of surjective endomorphisms. This strategy is in same taste and builds upon the work of Dani [8] on orbits of semisimple toral automorphisms.
Our setup has two players in which one of them (Alice) will be blocking open cylinder subsets of X P at every stage of a two-player game. To elaborate, one such cylinder is given by where x = (x 0 , . . . , x i , . . .) ∈ X P . For us, B(x i , r) will be always be the set of points in Q pi whose distance from x i is strictly less than r while B(x i , r) will also include those whose distance from x i is exactly r. We explain this game in § 3 after a brief tour of some of its older and related versions. Our aim is to prove the following statement in this paper: Theorem 1.1. Let Σ P be a solenoid over the circle S 1 and be any subset of affine surjective endomorphisms of Σ P such that 1. none of the A j 's is a non-trivial translation of Σ P , and 2. the collection of rational numbers {m j /n j } j∈N lying below the family {A j } belong to some finite ring extension of Z.
Then, there exists a cylinder absolute winning subset F ⊂ Σ P such that for any This is done in § 4. In the last section, we illustrate how this information can be used to infer something about the f -dimensional Hausdorff measure of F . We also discuss the strong game winning and incompressible nature of CAW subsets when P is finite.

Comparison with the work of Weil
It is plausible that some of our results given here may also be obtained from the more general framework discussed by Weil [23]. We take some time to expound the import of his main result.
Let (X, d) be a proper metric space (i. e. all closed balls are compact) and X a closed subset of X. Consider a family of subsets {R λ ⊂ X | λ ∈ Λ} which are called resonant sets and a family of contractions {ψ λ | ]0, 1] → 2 X | λ ∈ Λ}, indexed by some (same) countable set Λ. It is required that R λ ⊂ ψ λ (t 1 ) ⊂ ψ(t 2 ) for all 0 < t 1 < t 2 and λ. This datae is written in a concise form as F = (Λ, R λ , ψ λ ). The set of badly approximable points in S with respect to the family F is defined as Next, each R λ is assigned a height h λ with inf λ h λ > 0. The standard contraction ψ λ is then determined as ψ λ (c) := N c/h λ (R λ ), where N ε (S) denotes the set of all points of X in the ε-vicinity of elements of S. We further assume that our resonant sets R λ are nested with respect to the height function h, i. e., R λ ⊆ R β for every λ, β ∈ Λ such that h λ ≤ h β and that the values taken by h form a discrete subset of ] 0, ∞ [. For any collection S of subsets of X, the set X is said to be b * -diffuse with respect to S for some 0 < b * < 1 if there exists some r 0 > 0 such that for all balls B(x, r), x ∈ X, 0 < r < r 0 and S ∈ S, there exists a sub-ball The family F is locally contained in S if for any B(x, r), x ∈ X with r < r 0 and λ ∈ Λ with h λ ≤ 1/r, there is an S ∈ S such that B ∩ R λ ⊂ S. Concrete realizations of this abstract formalism include the case when X is the Euclidean space R n and S consists of all affine hyperplanes in X. Some examples of hyperplane diffuse sets are supports of absolutely decaying measures such as the Cantor sets and the Sierpiński triangle.
Theorem 1.2 (Weil, 2012). Let X ⊂ X be closed and b * -diffuse with respect to a collection S of subsets of X. Also, F is a family with nested resonant sets R λ and discrete heights, locally contained in S. The set BA X (F ) defined in (1.4) is Schmidt winning subset of X.
Many of the terms used above will be explained in § 3. Actually, it is shown in [23] that BA X (F ) is absolute winning with respect to S. This covers many important examples like the set of (s 1 , . . . , s n )-badly approximable vectors in R n , C 2 and Z 2 p , the set of sequences in the Bernoulli-shift which avoid all periodic sequences and the set of orbits of toral endomorphisms which stay away from periodic orbits. Curiously for us, we do not find any discussion on solenoidal endomorphisms in his work.
In order to be able to use his results, we would have to show that (a) the space X P is b * -diffuse with respect to the collection C of open cylinders in X P for an appropriate value of b * , and (b) the family of pre-images A −j B(y + z, t) is locally contained in C. We have endeavoured to provide a more direct proof here. In this sense, our work may be considered as an addition to the list of examples given in [23]. We must also point out that the assumption of having a Federer measure with X as its support in [23, Proposition 2.1] need not be true when X = X P , P is infinite and µ is the Haar measure on X P (cf. (2.8)).

Metric and measure structure on solenoids
Before we discuss the game, it is imperative that we say a few words about how balls in our space X P,n "look like." A metric on X P,n is given by where | · | is the usual Euclidean metric on R n and | · | p refers to the p-adic ultrametric on Q n p such that the diameter of (p −1 Z p ) n is p. Clearly, distance between any two distinct points of ∆(R n ) is at least 1 or in other words, the injectivity radius for the quotient map Π : X P,n → Σ P (n) equals 1. More generally, as per [5], a subset Z of any metric space X is said to be δ-uniformly discrete if the distance between any two distinct points of Z is at least δ. In this terminology, the set ∆(R n ) described above is 1-uniformly discrete in X P,n .
The definition of the metric in (2.1) ensures that balls B(x, r) ⊂ X P are direct products of their coordinatewise projections when (i) P is a finite set, or (ii) when r < 1. Moreover for r < 1, 1 ≤ p i r < p i for all i large enough. As the coordinates x i of x belong to Z pi for i ≫ 1, the projections B(x i , p i r) = Z pi for all but finitely many i's whereby we get that any such 'open' ball B(x, r) is actually open in X P while the 'closed' ball B(x, r) is a compact neighbourhood of x. It is easy to see that the diameter of B(x, r) is 2r but in most (all but one) of the places p ∈ P, the p-adic diameters of the projections will be strictly less than pr as distances in the non-archimedean fields are a discrete set. For a given r > 0, there might not be any integer j such that p j i = p i r. This seemingly minor issue is very crucial in our next set of calculations. If j i ∈ Z is such that p ji i ≤ r < p ji+1 i , then we call ⌊r⌋ i := p ji i . It should be noted that The lemma below may be of independent interest to the reader: Then, Proof. Let p i < x and m i ∈ N be the unique integer for which We know that m i = k if and only if x 1/(k+1) ≤ p i < x 1/k . This observation leads to decomposing P P (x) as a double product where ℓ ∈ N is such that x 1/(ℓ+1) ≤ 2 < x 1/ℓ (i. e., log x/ log 2 − 1 ≤ ℓ < log x/ log 2). Taking negative logarithms on both sides, we have θ P x 1/k − ℓ θ P x 1/(ℓ+1) + ln x .
When P is the full subset of primes, θ P ( x ) ∼ x as x → ∞. If P consists of all primes in an arithmetic progression with common difference k and (a, k) = 1 where a is one of the terms in the progression, then θ P ( x ) goes roughly as x/ϕ(k) as x → ∞. Here, ϕ stands for the Euler's totient function. For more on this, the reader is redirected to Montgomery and Vaughan's book [20]. We conclude that for all subsets P of primes which come from arithmetic progressions (and in particular the full subset), there exist some 0 < c 1 = c 1 (k) < 1 such that P P 1 r ≤ r c1 1/r ln(1/r) for all 0 < r ≪ 1.
This short exercise serves dual purpose. On one hand, it is a roundabout way of establishing the infiniteness of the Hausdorff dimension of X P when P is as above using the mass distribution principle. A finite non-zero measure µ whose support is a bounded subset of a metric space M is called a mass distribution on M . We require our dimension functions (also known as gauge functions) f : R ≥0 → R ≥0 to be increasing in some neighbourhood [0, a f ), continuous on (0, a f ), right continuous at 0 and f (r) = 0 iff r equals zero [11, pg. 33].
Proposition 2.2 (cf. [11]). Let µ be a mass distribution on a second countable metric space M such that for some dimension function f and δ 0 > 0, for some fixed c 2 > 0 and all subsets U with | U | ≤ δ 0 . Then, Now, let µ be the restriction to [0, 1] × p∈P Z p of the Haar measure ν on X P which is the product of the Haar measures on R and on each Q p , p ∈ P. We normalize it so that µ is a probability measure on X P . Any ball B(x, r) with radius 0 < r ≪ 1 will then have by (2.7) when P is the set of all primes in any infinite arithmetic progression with (a, k) = 1. After some more work, the above proposition will give us that the Hausdorff dimension of the space X P is infinite in such cases.
On the other hand, Lemma 2.1 will help us again in § 5 when we examine the dimension-theoretic largeness of various winning subsets of cylinder absolute games. In our version, Alice shall be dealing with the family C of closed subsets exactly one of whose co-ordinates x i , i ∈ {0, . . . , l − 1} is a fixed constant. The resulting ε-neighbourhood C(x, ε, i) of such a set P ∈ C as also defined in (1.3) will be called an (open) cylinder. Given a cylinder C = C(x, ε, i), we say that the radius of C is ε if i = 0 and the minimum such ε ′ for which C(x, ε ′ , i) = C(x, ε, i) otherwise. The index i is called the constraining coordinate of C. We emphasize that both Alice and Bob are fully aware of the radii of the balls chosen by the latter at any stage of our game by reading the real coordinate.

Infinite games on complete metric spaces
Let M be a complete metric space and F be a fixed subset of M . In the original game introduced by Schmidt [21], Alice and Bob are two players who each take turns to pick closed balls in M in the following manner: We have α, β ∈ (0, 1) to be two real numbers such that 1 − 2α + αβ > 0. The game begins with Bob choosing any closed ball B 0 = B(b 0 , r) ⊆ M subsequent to which Alice has to make a choice of some A 1 = B(a 1 , αr) such that A 1 ⊂ B 0 . After this, Bob picks B 1 = B(b 1 , βαr) ⊂ A 1 and the game goes on till infinity. We thus get a decreasing sequence of closed, non-empty subsets of a complete metric space Alice has a strategy to win the above game regardless of Bob's moves. Further, it is α -winning if it is (α, β) -winning for all β ∈ (0, 1) and Schmidt winning if it is α-winning for some α ∈ (0, 1).
For various applications of practical interest, one finds out that Alice need not bother herself too much about choosing the balls A j 's as long as she is able to block out neighbourhoods of certain undesirable points. This is true for example when M is the real line and F is the set BA of badly approximable numbers as discussed in [21] where Alice needs to be far from rational numbers with small denominators. Moreover if she is careful enough about her strategy, she has to worry about very few of such rationals -at times just one of them and hence, she need only shift the game outside of a ball B centered at some p/q ∈ Q ∩ B j at the j-th stage. This was formalized by McMullen [19] who called the new variant to be absolute winning games. Let β ∈ (0, 1

/3) and now Alice chooses open balls
Note that the countable intersection ∩ j B j might be bigger than a singleton set now and we have to set the winning condition to be ∩ j B j ∩ F = φ. The set F is said to be β-absolute winning if Alice has a strategy to win in this situation and it is called absolute winning if it is β-absolute winning for all β ∈ (0, 1/3).
In the same paper [19], McMullen also gave the concept of a strong winning set. We again have two parameters α, β ∈ ] 0, 1 [ but now Alice is allowed more options in the form of balls It continues to be mandatory that A i ⊂ B i−1 and B i ⊂ A i for all i. A subset F for which Alice has a winning strategy in this game is called an (α, β)-strong winning set. The subset F is said to be α-strong winning if it is (α, β)-strong winning for all β ∈ ] 0, 1 [ and strong winning if it is α-strong winning for some α > 0. For Euclidean spaces, a strong winning subset is Schmidt winning too and retains its strong winning property under quasisymmetric mappings [19, Theorem 1.2].
The absolute game has an obvious drawback that if F is the set of badly approximable vectors in R n for any n > 1, then Bob can force the game to be always centered on the hyperplane R n−1 × {0} and Alice is not able to win trivially. Therefore, it was proposed in [6] that she be allowed to block out a neighbourhood of some k -dimensional affine subspace of R n at each stage of the game. Taking this into consideration, they gave a family of games played on the Euclidean space R n called k-dimensional β-absolute games (0 < β < 1/3, 0 ≤ k < n) where Bob having chosen B 0 = B(b 0 , r 0 ) ⊂ R n , Alice picks some affine subspace V 1 of dimension k and for some 0 < ε 1 ≤ βr 1 removes the ε 1 -neighbourhood of V 1 , namely from B 0 . This is followed by Bob picking a closed ball B 1 ⊆ B 0 \ A 1 with radius (B 1 ) ≥ βr 0 and the game proceeds in a similar fashion. In general, the parameter ε j is allowed to depend on j subject only to 0 < ε j ≤ β · radius(B j ). Alice wins if ∩ j B j ∩ F = φ. As before, F ⊆ R n is k-dimensional β-absolute winning if Alice can win the k-dimensional β-absolute game over F irrespective of Bob's strategy. It is called k-dimensional absolute winning if it is k-dimensional β-absolute winning for all β ∈ (0, 1/3). It is clear from the definitions that for 0 ≤ k 1 < k 2 < n, if a set F ⊆ R n is k 1 -dimensional β-absolute winning, then it is k 2 -dimensional β-absolute winning too. Also, 0-dimensional β-absolute winning is the same as β-absolute winning.
All of this culminated in the axiomatization by Fishman, Simmons, and Urbański [12] where M is a complete metric space, H is a non-empty collection of closed subsets of M and F ⊆ M is fixed before the start of play. For 0 < β < 1, the set F is called (H, β)-absolute winning if Alice can ensure the intersection for some H j ∈ H and 0 < ε j ≤ β · radius (B j−1 ) at every j-th stage of the game. We follow Kleinbock and Ly [15] to declare Bob the winner by default if at any (finite) stage of the game, he is left with no legal choice of the ball B j to make. In the course of the game, Alice will have to make sure that such an event does not ever occur. This is keeping in mind the example of a Schmidt game illustrated in [16,Proposition 5.2] where Bob is not able to win because he has no option of B j left. The reader is cautioned at this point that in [12, Definition C.1], the authors resort to the opposite convention of Alice winning the game if it ends abruptly.
Ever since [21] came out, Schmidt games have been played and won over subsets of various metric spaces. We were unable to find any reasonable survey article covering the developments in the area. It will also be impossible to give here a comprehensive account of all the progress that has been made by different people and groups. We will have to contend ourselves by pointing to only a few representative works. Dani [7] formulated and proved results about the winning nature of the set of points in a homogeneous space G/Γ of a semisimple Lie group G whose orbits under a one-parameter subgroup action are bounded. Aravinda [2] showed that the set of points on any non-constant C 1 curve σ on the unit tangent sphere S p of any point p on a complete, non-compact Riemannian manifold M with constant negative curvature and finite Riemannian volume which lead to bounded geodesic orbits is Schmidt winning.
When Γ ⊂ G is an irreducible lattice of a connected, semisimple G with no compact factors, Kleinbock and Margulis [14] established that the subset of points in G/Γ with bounded H-orbits is of full Hausdorff dimension whenever H is a nonquasiunipotent one-parameter subgroup of G. Kleinbock and Weiss [16] allowed for Alice's and Bob's choices of subsets to be more flexible than just metric balls and used this to settle that the set of s-badly approximable vectors in R n is winning for any fixed s ∈ R n + . This was part of an effort to understand Schmidt's conjecture on the intersection of the sets of weighted badly approximable vectors for different weights which was finally resolved by Badziahin, Pollington, and Velani [3]. It was shown by Einsiedler, Ghosh, and Lytle [10] that the set of points on any C 1 curve which are badly approximable by rationals coming from a number field K is Schmidt winning. More generally, it is possible to define a hyperplane absolute game on any C 1 manifold. For example, it was recently proved in [1] that for any one parameter Ad-semisimple subsemigroup {g t } t≥0 of the product G of finitely many copies of SL 2 (R)'s, the set of points x belonging to any lattice quotient G/Γ of G and with bounded {g t }-orbit in G/Γ is hyperplane absolute winning.
In our setting, M shall be X P (or Σ P if you prefer), H is the family C of subsets described in § 2 and an example of the target set F is given below. A less contrived one will be available in the next section. A cylinder β-absolute game begins with Bob choosing a closed ball B 0 = B(x 0 , r 0 ). Subsequent to this, Alice blocks an open cylinder C 1 whose radius has to be less than or equal to βr 0 . The cylinders seem to us to be the appropriate replacement for the hyperplane neighbourhoods of [6] in metric spaces like solenoids. Recall that the exact value of the radius can be read off from the real coordinate. Moreover, radius (B j ) is required to be at least β · radius (B j−1 ) for all j ∈ N. The game of our interest goes as and F is said to be cylinder β-absolute winning if Alice can devise a method to win this game, i. e., j B j ∩ F = φ. It is cylinder absolute winning if there exists a 0 < β P = β P (F ) ≤ 1/3 such that F is cylinder β-absolute winning for all β ∈ ] 0, β P [. The supremum of such β P 's is christened the CAW dimension of F .
The basic idea of the proof remains the same as in [21,Theorem 2] and is being skipped here. We next give a theorem largely inspired by one of Dani [7].
Theorem 3.2. Let N be a countable indexing set and {A (n,t) ⊆ C (n,t) ⊂ X P | n ∈ N, t ∈ (0, 1)} be a family of set pairs where C (n,t) 's are restricted to be open cylinders in X P with the same fixed constraining coordinate i. If for any compact K ⊂ X P and µ ∈ (0, 1), there exist R ≥ 1, ε ∈ (0, 1) and a sequence (R n ) of positive reals with the following properties: 1. if n ∈ N and t ∈ (0, ε) are such that A (n,t) ∩ K = φ, then R n ≤ R and the radius r(C (n,t) ) of the cylinder C (n,t) is at most tR n , 2. if n 1 , n 2 ∈ N and t ∈ (0, ε) are such that both A (ni,t) intersect K nontrivially and the radius bounds of the associated cylinders are comparable, i. e., µR n1 ≤ R n2 ≤ µ −1 R n1 , then either n 1 = n 2 or d (A (n1,t) , A (n2,t) ) ≥ ε(R n1 + R n2 ).
Proof. Given any 0 < β < β 0 , let B 0 = B(x, r 0 ) be the initial closed ball of radius r 0 chosen by Bob to kick start the cylinder β-absolute game. Without loss of generality, we may assume that r 0 < 1/2 as well as that the balls B i chosen by Bob have radii r i → 0 (Alice can force this by removing some largest possible cylinder which is legally allowed at each turn). We let R, ε and (R n ) take the values dictated by our hypothesis for K = B 0 and µ = β 2 /2. Then, let k 0 ∈ N be the smallest such that µ k0 < min{εµr −1 0 , R −1 } and δ := µ k0+1 r 0 < ε. For k ≥ 1, mark h k → ∞ to be any strictly increasing subsequence such that βµ k r 0 < radius (B h k ) =: r k ≤ µ k r 0 . (

3.4)
This is well-defined as r k+1 ≥ βr k for all k ∈ N and µ k+1 < βµ k . We claim that Alice is able to play in such a manner that the closed ball B h k does not intersect any A (n,δ) with R n ≥ µ k−k0 . The limit point b ∞ = ∩ ∞ k=0 B k = ∩ ∞ k=0 B h k shall then be in F and the proof of the theorem will be done (as β < β 0 is arbitrary).
Our claim is vacuously true for k = 0 as R n ≤ R < µ −k0 for all A (n,δ) intersecting B 0 non-trivially by our assumption. Thereafter, supposing that the claim holds for k, we show it to be true for k + 1. Since the sets A (n,δ) with the corresponding cylinder radii bounds R n ≥ µ k−k0 have already been taken care of, we only need to show that Alice can now ensure B h k+1 does not intersect A (n,δ) for any n ∈ N such that µ k+1−k0 ≤ R n < µ k−k0 . As hinted before, she has to worry about exactly one such subset. For, if both , then the second condition of the theorem says that d(A (n1,δ) , A (n2,δ) ) ≥ ε(R n1 + R n2 ) ≥ 2εµ k+1−k0 while |B h k | ≤ 2r k ≤ 2µ k r 0 and we have a contradiction.
If n ∈ N is the unique index for which to be the open cylinder C (n,δ) and since this constitutes a legal move. It only remains to be argued that Bob has some choice of B h k +1 ⊂ B h k \ C h k +1 left (in fact, plenty of them). If the constraining coordinate i of C h k +1 equals zero, we only need to find a point in the closed ball B x k,0 , (1 − β)r k ⊂ R which is at a Euclidean distance βr k from some open ball B(y, βr k ) containing the projection π 0 (C h k +1 ) of C (n,δ) in the archimedean coordinate. This is clearly possible as long as β < 1/3.
Else if i > 0, let B(x k,i , ⌊p i r k ⌋ i ), B(y, p i r ′ k+1 ) ⊂ Q pi be the respective images of B h k and C h k +1 under the projection π i onto the i-th coordinate. As B h k ∩C h k +1 = φ by our assumption, we get that B(x k,i , ⌊p i r k ⌋ i ) ∩ B(y, r ′ k+1 ) = φ too. Being balls in an ultrametric space, one of them then has to be contained in the other and because β < 1/p i , we have and thereby B(y, p i r ′ k+1 ) B(x k,i , p i ⌊r k ⌋ i ) = B(y, p i ⌊r k ⌋ i ). Bob picks a point z i ∈ B(y, p i ⌊r k ⌋ i ) whose distance from y is equal to p i ⌊r k ⌋ i and if where π ⊥ i : X → R × ′ j =i Q pj is the complementary projection of π i , he defines This has diameter 2βr k , is contained in B h k and avoids the cylinder C h k +1 .
Given any positive lower bound on the CAW dimension, we can boost it to an absolute quantity (depending on P alone) for finite solenoids. Lemma 3.3. Let P = {p 1 < · · · < p l−1 } be finite. Any CAW subset S of X P has winning dimension ≥ min β P := {1/3, 1/p l−1 }.
Proof. Assume 0 < β < β P . We are guaranteed that there exists some 0 < β ′ < β such that S is cylinder β ′ -absolute winning. Changing the game parameter from β ′ to β only enlarges the set of choices available to Alice while Bob continues to have some legal choice left as long as β < 1/3 and β < 1/p l−1 ≤ 1/p i , if i > 0 is the constraining coordinate of the cylinder blocked by Alice in the previous move. Also, all of his valid moves in the cylinder β-absolute game remain so in the β ′ -game. Alice just needs to pretend that the game parameter is β ′ and follow her winning strategy for the same.

Non-dense orbits of solenoidal maps
As already mentioned in § 1, an affine endomorphism A : Σ P → Σ P is of the form x → (m/n)x + a for some m/n ∈ R and a ∈ Σ P . Here, R is the set of endomorphisms of the solenoid Σ P = XP /∆(R) given by the ring R = Z {1/p i | p i ∈ P} . The affine transformation A is invertible iff n/m ∈ R too.
Next, the cylinder β-absolute game on Σ P can be shifted to a game played on X P once the radii of the balls B i ⊂ Σ P become small enough (say < 1/2). This can be forced on Bob in finitely many steps after the beginning of the game.
Pick some y ∈ Σ P and let A be any fixed affine transformation of Σ P with its linear part m/n ∈ R \ {0, ±1}. We abuse notation and call any of its lifts from X P → X P to be A too. Note that any such lift is an invertible self map of X P as long as A : Σ P → Σ P is surjective. Further, let F A (y) denote the set of points x ∈ X P whose image x := Π(x) has its A-orbit not entering some δ(x, y, A)-neighbourhood of y. The goal for Alice is to avoid ε-neighbourhoods of the grid points Π −1 ({y}) = y + ∆(R) (for some y ∈ Π −1 ({y})) which are all at least a unit distance away from each other. Otherwise said, the set F A (y) that Alice should aim for is and FA(y) /∆(R) shall be the image set for the game played on Σ P . By our assumption about A, there exists i ≥ 0 such that | m /n| pi =: λ i > 1. We let λ A to be sup i λ i . This is finite, attained for some i = i 0 and strictly greater than 1. In particular for any x 1 , x 2 ∈ X P , we have that as translation by any a is an isometry of Σ P (and X P ). Equivalently for any two subsets F 1 , F 2 ⊂ X P , The constraining coordinate of all the cylinders removed by Alice will be some fixed i 0 for which λ i0 = λ A . Let 0 < µ < 1 and ℓ ∈ N be the smallest for which λ −ℓ A < µ. If a = m −ℓ , then ( m /n) −j R ⊆ aR for all j ∈ {0, . . . , ℓ}. Not unlike R, the points of aR too constitute a δ-uniformly discrete set for some 0 < δ = δ(a, P) ≤ 1. We also choose b ≥ 1 given by and let This belongs to ] 0, 1/3 ] and thereby for any z 1 , z 2 ∈ ∆(R) such that y + z 1 = A −j (y + z 2 ) for some 0 ≤ j ≤ ℓ, we have If j 1 ≤ j 2 are any two exponents such that µ k+1 < λ −j2 A ≤ µ k for some k ∈ N, then j 2 − j 1 ≤ ℓ by the very definition of ℓ. Hence, for 0 < t < µt 0 /2 and any z 1 , z 2 ∈ ∆(R) for which y + z 1 = A −(j2−j1) (y + z 2 ), we get that be the countable indexing set in our Theorem 3.2. The first hypothesis therein is satisfied by taking R = 1 and for n = (j, y + z), letting which suggests that we should take R n = λ −j A . Clearly, the second hypothesis has been shown to hold here in (4.7). Hence, we infer that F A (y) = ∪ t>0 X P \ ∞ j=0 A −j R + B(y, t) is a CAW subset of X P with winning dimension as in the statement of Theorem 3.2 and so is its image in Σ P . Because Proposition 3.1, we can extend this result to the set of points whose A-orbits avoid some neighbourhoods of countably many points {y k } k∈N ⊂ Σ P .
If A(x) = (m/n)x + a is such that m/n = −1, then A 2 is the identity endomorphism. In this case, Alice only needs to move the game away from the countable set {y k }∪{a−y k }. This is trivial. The situation is even simpler when A is just the identity map. Now, let Y be the set consisting of all those points of Σ P which have a periodic orbit for some B ∈ R \ {±1}. This is countable and leads us to conclude: x + a j } j∈N be any subset of affine surjective endomorphisms of the solenoid Σ P such that 1. none of the A j 's is a non-trivial translation of Σ P , and 2. the collection of rational numbers {m j /n j } j∈N belong to some finite extension of Z.
Then, the set of points whose orbit closure under the action of any of the A j 's does not contain any periodic B-orbit for all B ∈ R \ {±1} is cylinder absolute winning.
Proof. If {A j } ⊂ Z {1/p 1 , . . . , 1/p n | p i ∈ P} , then the winning dimension of each of the subsets F Aj is at least min{1/3, min{1/p i | 1 ≤ i ≤ n}} > 0. This is also a lower bound on the CAW dimension of the intersection ∩ j F Aj invoking Proposition 3.1 once again.
Note that even though R is a countable set, we cannot further this argument to take intersections over any arbitrarily chosen sequences of affine surjective endomorphisms of Σ P . This is because the lower bound on the winning dimension of the CAW subsets of Σ P corresponding to each A is dependent on A itself in terms of i 0 for which λ i0 = λ A . However, for finite P, each such β 0 is at least min{1/3, min{1/p i | p i ∈ P}}. We can then remove the second condition in Theorem 4.1 to get Theorem 4.2. Let P be a finite set of rational primes and {A j } be any sequence of affine surjective endomorphisms of Σ P such that none of the A j 's is a translation. The set of points whose orbit closure under the action of any A j does not contain any periodic B-orbit for all B ∈ R\{±1} is CAW with winning dimension at least min{1/3, 1/p l−1 }. Here, p l−1 is the largest prime in P.
In particular, this is true of the collection of all surjective endomorphisms of Σ P .

Sizes of CAW subsets
Let P be finite. We start by discussing the implications of CAW property of a subset F for a strong game played on X P with F as its target.
Proposition 5.1. A CAW subset of X P is α-strong winning for all α < β P .
Proof. Without loss of generality, we may take the CAW dimension of F to be β P due to Lemma 3.3. This means our target set F is β-CAW for all 0 < β < β P . Now, suppose that α ∈ ] 0, β P [ and γ ∈ ] 0, 1 [ are any fixed (strong) game parameters for Alice and Bob, respectively.
Given a ball B 0 = B(x, r) chosen by Bob at any stage of the strong game, Alice checks the cylinder C with radius ( C ) ≤ αγr to be removed by her in accordance with her winning strategy for F when playing the cylinder (αγ)absolute game. If B ∩ C = φ, she chooses any A ⊂ B allowed by the rules of the strong game. Assume this to not be the case for the rest of this proof.
If the constraining coordinate i of C is archimedean, Alice has no problem in choosing a Euclidean ball A ⊂ π 0 (B) \ π 0 (C) with radius ( A ) ≥ αr as α ≤ β P ≤ 1/3. Else again when i > 0, we have π i (C) π i (B) because We can moreover take the center x i of π i (B) to be the same as that of π i (C). Let z i ∈ π i (B) \ π i (C). Then, | z i − x i | pi = p i ⌊ r ⌋ i and the ultrametric also gives us that B Qp i (z i , ⌊ r ⌋ i ) ⊂ π i (B) \ π i (C). In either case, the pre-images π −1 0 (A) or π −1 i B Qp i (z i , ⌊ r ⌋ i ) contain a ball of X P of radius at least β P r ≥ αr which lies inside B \ C. Alice chooses one such A 1 to be her next move. Bob's choice of any B 1 ⊂ A 1 with radius ( B 1 ) ≥ γ · radius ( A 1 ) ≥ αγr immediately after is also a valid move in cylinder (αγ)-absolute game.
It should be mentioned here that the relationship between winning sets for strong games and quasisymmetric homeomorphisms of X P is not clear to us. Nor do we have the analogous statement of Proposition 5.1 for the full solenoid.

Incompressibility
The next result is about the incompressible behaviour of cylinder absolute winning subsets of X P . A set S ⊂ X P is strongly affinely incompressible if for any non-empty open subset U and any sequence of invertible affine homomoprhisms (Ψ i ) i∈N , the set ∩ i∈N Ψ −1 i S ∩ U has the same Hausdorff dimension as U [6,9]. It is our claim that CAW subsets of X P are strongly affinely incompressible for finite P. We show this by proving a lower bound on the CAW dimension of Ψ −1 S ∩ U in terms of windim S for any affine map Ψ of X P . Together with Proposition 3.1, this will give us that the intersection of any countably many pre-images of a CAW subset under invertible affine homomorphisms is CAW too.
Theorem 5.2. Let P be finite, U ⊂ X P open and S be any CAW subset. Also, let Ψ : X P → X P be an invertible affine homomorphism. Then, the set Ψ −1 S ∪ (X P \ U ) is also CAW with winning dimension at least β P .
Proof. As in Proposition 5.1, we may take the CAW dimension of S to be β P without loss of generality. Let us first make some reductions to simpler situations. If the diameters | B k | of balls chosen by Bob don't go to zero as k → ∞, then ∩ k B k contains an open ball inside it. As S is a winning subset, it has to be dense and in turn its pre-image Ψ −1 S is also dense in X P . Second, if B k ∩ (X P \ U ) = φ for infinitely many k, then they form a decreasing sequence of closed subsets of the compact ball B 1 . Their intersection cannot be empty and hence ∩ k B k contains a point of X P \ U resulting in Alice's victory. It is safe to exclude both of these events from the rest of the proof. We can moreover take that B 0 ⊂ U .
Let 0 < β < β P and Ψ(x) = Dx + a as explained before. Following [6], Alice will run a 'hypothetical' Game 2 (in her mind) where the target set is S and a different game parameter β ′ which is some positive power of β. She carefully decides and projects some of Bob's moves in the Ψ −1 (S) ∪ (X P \ U ), β -game to construct choices made by a hypothetical Bob II in Game 2. Since there is a winning strategy for the latter by hypothesis, she channels the winning moves in this second game via the inverse map Ψ −1 to win over Ψ −1 (S). We take which makes sense as D is a rational number. As D = 0, we have 1 ≤ λ Ψ < ∞ for any Ψ and any P. It is clear that as d is a translation-invariant metric on X P . We re-label the choices made by Bob such that radius ( B 0 ) < 1/λ Ψ . Let n ∈ N be the smallest positive natural number for which λ Ψ λ Ψ −1 (β + 1)β n−2 < 1, β ′ = β n and η := (β + 1)β n−1 .

(5.4)
Alice waits for the stages 0 = j 1 < j 2 < · · · in the original Ψ −1 (S) ∪ (X P \ U ), β -cylinder absolute game when for the first time. Notice that this is well-defined and exists because we assumed | B k | → 0 and the radius of Bob's choice at (k + 1)-th step cannot shrink by a factor of more than β compared to that of his choice at the k-th step for all k. Imitating [6], denote to be blocked next in the β-cylinder absolute game with target set Ψ −1 (S). By design, λ Ψ λ Ψ −1 η < β and it only remains to show that Bob has some As we have seen in the proof of Theorem 3.2, this is clearly not a problem as β < β P ≤ min{1/3, 1/p i k+1 } when i k+1 > 0 or otherwise. All of Bob's subsequent choices in Game 1, including B j k+1 = B(x k+1 , r k+1 ), obey | x k,0 − x k+1,0 | ≤ r k − r k+1 (5.8) in the archimedean coordinate and Then, The conclusion cannot be escaped that the corresponding ball which gives that ) by Alice's choice of marking for B j k+1 . The computations are not very different when the constraining coordinate of C ′ k+1 is archimedean.
For general P, we are only able to show the largeness of countable intersections of pre-images under translations of X P . Proposition 5.3. Let (a k ) k∈N ⊂ X P be any arbitrary sequence and S be a CAW subset with winning dimension β 0 . Then, so is S ∩ k∈N (S + a k ).
Proof. Each of the translations Ψ k (x) := x−a k is an isometry and in particular, does not change the shape of the balls in X P . Given any such single Ψ, we argue that Ψ −1 (S) has the same CAW dimension as S. Alice simply translates back her choices for the Game 2 described above by −a when Ψ(x) = x + a and projects Bob's succeeding choice forward by Ψ. Note that as λ Ψ k = λ Ψ −1 k = 1 for all k, she should take β ′ = β for any 0 < β < β 0 . One should also replace η = 1 in the previous calculations. The countable intersection property then follows by Proposition 3.1.

Hausdorff dimension and measures
Lastly, we will try to understand the sizes of CAW sets in terms of Hausdorff dimensions and measures. Towards this goal, we will require an estimate on the number of legal choices that Bob has at any stage of the game.
Lemma 5.4. Let 0 < β ≪ 1. Then, the maximum number of pairwise disjoint balls of radius βr contained in any closed ball B(x, r) ⊂ X P = R × ′ j>0 Q pj which do not intersect an open cylinder C(y, βr, i) and also maintain a distance at least βr from each other is given by where θ P ( t ) = p∈P, p≤t ln p.
Proof. Because (2.1), every closed ball is the Cartesian product of its coordinatewise projections. In any non-archimedean coordinate j such that p j ≤ 1/β, there are at least (p j ⌊ β ⌋ j ) −1 pairwise disjoint balls contained in the projection π j B(x, r) = B(x j , ⌊p j r⌋ j ) whose radius equals ⌊p j βr⌋ j . Each of them also maintain a distance of at least p 2 j ⌊ βr ⌋ j from each other which means that the pre-images of any two such sub-balls in X P are ≥ p j ⌊ βr ⌋ j > βr away. The lower bound (p j ⌊ β ⌋ j ) −1 equals ⌊ 1/β ⌋ j unless β is an integral power of p j in which case it is p −1 j ⌊ 1/β ⌋ j . Note that this can happen for at most one prime for any given β and we have already assumed β < 1/p j . In the real coordinate, this number is ≫ β −1 even when we ask that the balls are at least βr apart. The pre-images under the projection map π β : X P → R × pj ≤1/β Q pj of any product of these sub-balls of π j B(x, r) for j = 0 or p j ≤ 1/β are pairwise disjoint, each contain at least one sub-ball of B(x, r) of radius βr and the minimum distance between any two of those pre-images is ≥ βr.
When asking for only those sub-balls that do not intersect the open cylinder C(y, βr, i), it is necessary and sufficient that we restrict ourselves to only those from the above chosen collection whose images in π i QB − (x, r) do not intersect π i C(y, βr, i) . Otherwise said, all coordinates but i are not affected. If i = 0, the number of such balls in π 0 QB − (x, r) = B(x 0 , r) that do not intersect π 0 C(y, βr, 0) = B(y 0 , βr) is still ≫ β −1 albeit with a smaller constant. Else, it is at least (p i β) −1 − 1 > β −1 /2p i . The Cartesian product of these disjoint balls in Q pj 's and R then gives us that by a calculation similar to the one in Lemma 2.1.
As discussed in § 2, θ P ( t ) ∼ t when t → ∞ and P is the full set of primes. More generally, it shows an asymptotic linear growth with t when P is any (infinite) set of all primes in an arithmetic progression. We record that the constant implied by the Vinogradov notation in (5.13) is independent of the constraining coordinate of the cylinder C. For finite P, we use a slightly different lower bound.
Lemma 5.5. Let | P | = l − 1 and p l−1 be the largest prime in P. Then, where the implied constant may depend on the primes p 1 , . . . , p l−1 and l.
Proof. We only need to replace the lower bound for the number of disjoint sub-balls in each non-archimedean coordinate by β −1 /p j and the rest of the argument remains the same.
Suppose F is a cylinder absolute winning subset of X and that Alice always plays according to a winning strategy if it is available. We shall now construct a subset F * ⊆ F which corresponds to the points obtained when Bob is only to allowed to choose one of the N C (β)-many sub-balls described in Lemmata 5.4 or 5.5 at each stage of the game. This resembles closely a device from Kristensen [17] (see also [21,Theorem 6]). Proposition 5.6. Let f be any dimension function such that Then, the f -dimensional Hausdorff measure of any CAW subset of X P is greater than zero.
Proof. Let β 0 , δ 0 > 0 be small enough so that β 0 is less than the winning dimension of our CAW set F , δ 0 < 1 and They exist by virtue of our hypothesis about f . Now, let Λ := {0, 1, . . . , N C (β 0 )− 1} N , the sequence space each of whose element λ = (λ k ) k∈N corresponds to a sequence of choices made by Bob when he is only allowed to choose from one of the N C (β 0 )-many disjoint sub-balls inside B k−1 . The choices made by him at the k-th stage are labelled B(λ 1 , λ 2 , . . . , λ k ). If λ = λ ′ , they differ in some entry k 0 and the corresponding balls B(λ 1 , . . . , λ k0 ) and B(λ ′ 1 , . . . , λ ′ k0 ) are disjoint. This implies that the points obtained at infinity, a ∞ (λ) = ∩ k→∞ B(λ 1 , . . . , λ k ) = a ∞ (λ ′ ) = ∩ k→∞ B(λ ′ 1 , . . . , λ ′ k ) ∈ F under the belief that Alice is following a winning strategy for the target set F with game parameter β 0 . Let We show that H f (F * ) > 0 and this shall in turn give us our desired statement. For this, the space F * is mapped in a continuous fashion (via the bijection with Λ) onto [0, 1] using the N C (β 0 )-adic expansion of real numbers, namely a ∞ (λ) → 0.λ 1 λ 2 · · · . Call this map ψ and let (U n ) n∈N be any δ-cover of F * for some δ < δ 0 . Without loss of generality, let U n ⊂ F * for all n. Plainly, ψ(U n ) n∈N is a cover for [0, 1] and since diameter is an outer measure on R, we get that 1 ≤ n∈N | ψ(U n ) | . so that j n > 0 for all | U n | small enough and furthermore, | U n | < β jn 0 . Thus, U n intersects non-trivially with at most one of the balls B(λ 1 , . . . , λ jn ) as any two such are at least β jn 0 apart. Further, being a subset of F * it is completely contained inside some B(λ 1 , . . . , λ jn ). The latter itself is mapped by ψ into the interval of length N C (β 0 ) −jn of I consisting of numbers whose N C (β 0 )-adic expansion begins with 0.λ 1 · · · λ jn . We conclude that | ψ(U n ) | ≤ N C (β 0 ) −jn and thereby, As | U n | ≤ δ < δ 0 < 1 for all n ∈ N and by our assumption, we have that n∈N f (| U n |) ≥ N C (β 0 ) −1 2 log NC(β0)/ log β0 (5.20) for any arbitrary δ-cover (U n ) of F * . Thus, the infimum of the sums on the left side of (5.20) taken over all δ-coverings of F * is a positive number independent of δ. Letting δ → 0 from the right, we conclude that the f -dimensional Hausdorff measure of F * is strictly positive. In particular, this proves our claim.
Corollary 5.7. Let Σ be the full solenoid over S 1 . Then, the Hausdorff dimension of any CAW subset of Σ is infinite.
Proof. We know N C (β) rises faster than β c/(β log β) as β → 0 for some absolute constant c > 0, when P is the set of all rational primes. Take f to be the power function r → r n for some n ∈ N. The condition in Proposition 5.6 is satisfied then and we get that the n-dimensional Hausdorff measure of any CAW subset of Σ is positive. Finally, we let n → ∞.
The same is true for CAW subsets of Σ P , when P is an infinite set consisting of all primes in some arithmetic progression. It will also be interesting to study the class of exact dimension functions for the spaces X P and their CAW subsets [11]. For finite P, we are satisfied with a statement about maximality of Hausdorff dimension.
Proposition 5.8. Let | P | < ∞ and F ⊂ X P be a β-CAW set. Then, Proof. The proof is the same as that for Proposition 5.6 till (5.18) with β replacing β 0 everywhere.
Once again if we let β → 0 for a CAW set, we get Corollary 5.9. Any CAW subset of X P with | P | = l − 1 has Hausdorff dimension equal to l. In particular, the collection of points described in Cororllary 4.2 has full dimension.
Remark. The proofs of Propositions 5.6 and 5.8 are suggestive that while there can be Schmidt winning subsets in the example metric space of Kleinbock and Weiss [16] mentioned in § 3 which are of Hausdorff dimension zero (in fact, countable), any absolute winning subset thereof shall have to be of full dimension.