EQUIDISTRIBUTION WITH AN ERROR RATE AND DIOPHANTINE APPROXIMATION OVER A LOCAL FIELD OF POSITIVE CHARACTERISTIC

. For a local ﬁeld K of formal Laurent series and its ring Z of polynomials, we prove a pointwise equidistribution with an error rate of each H -orbit in SL ( d, K ) /SL ( d, Z ) for a certain proper subgroup H of a horospherical group, extending a work of Kleinbock-Shi-Weiss. We obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson- Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class C α .

1. Introduction. Let K be the field F q ((t −1 )) of formal Laurent series in t −1 over a finite field F q of order q and let Z be the ring F q [t] of polynomials in t over F q . The absolute value | · | on K is given by |f | = q deg(f ) . Let O be the ring of formal power series F q [[t −1 ]] and λ be the Haar measure on K normalized by λ(O) = 1. For m, n ∈ N and d = m + n, let G = SL(d, K) and Γ = SL(d, Z). Let us denote by a + the set of d-tuples a = (a 1 , · · · , a d ) ∈ N d such that Given a = (a 1 , · · · , a d ) ∈ a + , let us define g a = diag(t a1 , · · · , t am , t −am+1 , · · · , t −am+n ).
In this paper, we study the action of the subsemigroup of G generated by g a on the space X = G/Γ. Since g a -action on X is ergodic with respect to the Haar probability measure µ on X, for µ-almost every x ∈ X, we have We are interested in the following question: given a proper subgroup L of G, does the above convergence of Birkhoff average of g a -translates still hold for almost every point in an L-orbit? This question was studied in [8] and [17] for a real Lie group G and a lattice subgroup Γ of G.
When L is the horospherical subgroup H + = {u ∈ G|g −n a ug n a → e as n → ∞} of G, then the answer is affirmative (cf. [11]). In this article, we consider a subgroup H of H + which has applications in metric Diophantine approximation (Propsition 1.2 and Theorem 1.4 below). Since H Mat m×n (K), the Haar measure on H is isomorphic to λ mn , which we will denote by λ H . Let us also denote the Haar measure on G such that λ G (K) = 1 for a maximal compact subgroup K by λ G . We first show that for compactly supported smooth functions, pointwise ergodic theorem for g a -action holds for almost every point in each H-orbit. A function f : G → R is smooth if there is a compact open subgroup U ⊂ G such that f is Uinvariant, i.e. it is locally constant. Let us denote by C ∞ c (X) the set of compactly supported functions on X = G/Γ whose lifts on G are Γ-invariant smooth functions on G. To show a similar result for functions with non-compact support, we make use of lattice point counting results. For p ∈ Z n , q ∈ Z m and A ∈ Mat m×n (K), consider the inequality Aq − p ∞ < φ( q ∞ ), 1 ≤ q ∞ ≤ q T (2) where · ∞ is the supremum norm. It was shown in [3] that the number of vectors (p, q) ∈ Z n × Z m satisfying the inequality (2) is for λ mn -almost every A ∈ Mat m×n (K), where Φ(T ) = q∈Z, q ∞ ≤q T φ( q ∞ ) n .
Generalizing this result, we prove the following weighted version. Consider the weighted quasi-norms given by and y β = max 1≤j≤n |y j | 1/aj+m for x ∈ K m and y ∈ K n . Denote by N R (T, A) the number of nonzero vectors (p, q) ∈ Z m × Z n satisfying the following Diophantine inequalities for λ mn -almost every A ∈ Mat m×n (K).
For the precise formula of Ψ R , see Theorem 4.1. Using Proposition 1.2, we obtain an ergodic theorem for certain unbounded functions as appeared in [5] and [8]. Every Z-submodule ∆ of rank r in K d has a Zbasis. For L = K∆, we denote by vol(L/∆) the covolume of ∆ in L, normalized by vol(K r /Z r ) = 1 for any r ∈ N. See for example [9] for a detailed definition.
In particular, we call a Z-submodule Λ of rank d a Z-lattice and a Z-lattice is unimodular if its covolume in K d is 1. Since SL(d, K) acts transitively on the space of unimodular lattices in K d and the stabilizer of Z d is SL(d, Z), we can identify the space with X = SL(d, K)/SL(d, Z). For a unimodular lattice Λ ⊂ K d and a Z-submodule ∆ ≤ Λ, let us define Let C α (X) be the space of functions ϕ on X such that (1) The function ϕ : X → R is continuous except on a µ-null set, and (2) There exists C > 0 such that for all Λ ∈ X, we have |ϕ(Λ)| ≤ Cα(Λ).
Theorem 1.4 (Counting with directions). Let C 1 and C 2 are measurable subsets of S m−1 and S n−1 , respectively, with measure zero boundaries with respect to λ S m−1 , λ S m−1 , respectively. As T → ∞, holds for λ mn -almost every A ∈ Mat m×n (K).
The structure of this paper is as follows: Section 2 is devoted to prove the uniform quantitative non-divergence of H-orbits. In Section 3, we prove an effective double equidistribution and use it to obtain the pointwise equidistribution for compactly supported continuous functions. We give an asymptotic formula with an error rate for the number of solutions to the Diophantine approximation with weights in Section 4. Using this result, we show in Section 5 the pointwise equidistribution for unbounded functions contained in C α (X).
2. Quantitative non-divergence. Throughout the article, we will mostly use the supremum norm For each Z-submodule Λ ⊂ K d of rank r, we can choose a basis v 1 , v 2 , · · · , v r and define the norm · by This norm is independent of the choice of the basis.
Also, let δ : X → R + be the length of shortest vector, i.e.
If δ(gΓ) < , then there exists a non-zero vector w ∈ Z d such that gw ∞ < .  such that every polynomial f ∈ K[x 1 , · · · , x r ] of degree deg(f ) ≤ s is (C, 1 rs )-good on K r with respect to the Haar measure λ r on K r . Let X = {Λ ∈ X | δ(Λ) ≥ }. By Mahler's compactness criterion, it follows that X is compact. (2) sup B ψ ∆ ≥ ρ for every ∆.
Then for any 0 < < ρ, we have where λ l is the Haar measure on K l .
Let us first check the condition (2) of Theorem 2.3 for h(A) = g a u A g, where u A is as in (1). Remark that the lower bound in condition (2) is expressed in terms of the minimum of a i 's. Denote min a := min  Proof. We follow the idea of Lemma 5.1 in [16]. Let σ : G → GL(V ) be the exterior The H-invariant subspace W can be described as follows: if r ≤ m, then and if r > m, then Let {f kl } 1≤k≤m,1≤l≤n be the standard basis of M . From the definition, we have By injectivity of S, there exists c 0 > 0 such that For a vector v in V given by a e 1 ∧ e 2 + b e 1 ∧ e 3 + c e 2 ∧ e 3 , we have Corollary 2.6. Let L be a compact subset of X and B ⊂ H be a ball centered at e ⊂ H. Then there exists N = N (B, L) such that for every 0 < < 1, any x = gΓ ∈ L, and any a ∈ a + with min a ≥ N we have where the implied constant depends on B and L.
is positive, say c 1 . Together with Lemma 2.4, we can find c > 0 such that for any g ∈ π −1 (L) we have sup A∈B g a u A gv ∞ ≥ cq min a gv ∞ ≥ cq N c 1 . Thus, the conditions of Theorem 2.3 are satisfied with ρ = cq N c 1 .

3.
Pointwise equidistribution with an error rate. In this section, we prove Theorem 1.1 which gives the pointwise equidistribution for compactly supported functions in each H-orbit. Before proving Theorem 1.1, we first prove effective equidistribution of g a -translates of H-orbits, a positive characteristic analog of a theorem of Kleinbock-Margulis [7]. For the case of equal weights, i.e. when a = (a, · · · , a, b, · · · , b) with ma = nb, see also [11]. For a given φ ∈ L 2 (X), we denote its L 2 -norm by φ 2 . Let K be the maximal compact subgroup SL(n, O) of G and K (l) be the l-th congruence subgroup (Id + t −l Mat(n, O)) ∩ K of K. Following [4] for the adelic case (and Rühr's thesis [15] for the p-adic case), let us define the degree-k Sobolev norm S k (φ) of φ as follows. Let Av l be the averaging projection to the set of K (l) -invariant functions, Let pr[l] be the l-th level projection Av l − Av l−1 for l ≥ 1 and pr[0] = Av 0 . Define For example, we have The following properties of the Sobolev norm will be used in the sequel.
The key idea of the proof is to use Cauchy-Schwartz inequality, after decomposing φ as l≥0 pr[l]φ. We omit the details of the proof, which is analogous to [4] and [15].
Using arguments of [4], one can easily obtain the following effective decay of matrix coefficients of the regular representation.
holds for some C 0 > 0 which depends only on the group G and Γ.
Proof. It follows from Theorem 2.1 of [1] (see also [13] for the setting more general than the function field case) that there exists δ 0 > 0 such that holds for some C > 0 depending on the smoothness of the functions φ and ψ, i.e. the choice of compact subgroup U in the definition of the smooth function. By replacing the L 2 -norm above by a Sobolev norm of degree d 2 which is equal to dim G + 1, we apply the argument of subsection A.8 of [4] (or Proposition 3.2.1 of [15]) to conclude that the implied constant in the statement of the proposition depends only on G and Γ.
Using Proposition 3.2, we prove the effective equidistribution of expanding translates of H-orbits. In the proof of the theorem, we will use the decomposition of G into where H − = Im 0 L In : L ∈ Mat n×m (K) and H 0 = P 0 0 Q : P ∈ GL(m, K), Q ∈ GL(n, K), det(P ) det(Q) = 1 . It is easy to check that the decomposition is unique, more precisely, Before proving the effective equidistribution for general case, we present the proof of the following special case.
with the same δ 0 as in Proposition 3.2.
Proof. Note that there is a level-l congruence subgroup K (l) of K such that f and φ are invariant under K (l) ∩ H and K (l) , respectively. Taking l large enough, we may assume that K (l) ⊂ B G ( r 2 ). Choose , which is supported on the small neighborhood of x ∈ X. This definition makes sense because π x is injective on B G (2r). Since the modular function on Therefore, Using the above proposition, we will prove the general case. Let us define the integral vector a of equal weight by a := (min a)e where e = (n, . . . , n, m . . . , m). Note that for i such that − 1 ≤ i 2(m+n) < for some ∈ N, ai − a ∈ a + and min(ai − a ) ≥ min ai/2. Theorem 3.4 (Effective equidistribution of expanding translates). There exists δ 1 > 0 such that the following holds. For any f ∈ C ∞ c (H), φ ∈ C ∞ c (X) and for any compact L ⊂ X, there exists C 1 = C 1 (f, φ, L) such that for all x ∈ L and i ∈ N, we have Proof. By choosing large enough C 1 , it is enough to show the proposition for large enough i. The proof is analogous to the argument of [7]. Throughout the proof, we denote by B H (r), B G (r) the ball of radius r centered at e in H, G, respectively. Without loss of generality, we may assume that X φ dµ = 0 and φ is invariant under an open compact subgroup K φ of G. Let κ > 0 be small enough so that containing the support of f and enlarge the ball B slightly to obtain B = B H (r 1 + q −2 min(ai−a ) q −κi ).
We choose i 0 large enough so that for all i ≥ i 0 , we have λ H ( B) ≤ 2λ H (B) and min(ai − a ) ≥ N ( B, L) for the constant N ( B, L) of Corollary 2.6. It follows that given any > 0, any x ∈ L and V 1 := {h ∈ B | δ(g ai−a hx) < }, we have λ H (V 1 ) 1/(mn(m+n)) λ H ( B) ≤ 2 1/(mn(m+n)) λ H (B). Therefore it holds that for the degree-d 2 Sobolev norm by Lemma 3.1 (1). Now let us compute the integral on the complement V 2 := B\V 1 . Given any y ∈ B H (q −κi ), let k y = g −1 ai−a yg ai−a ∈ H. Since d(e, k y ) ≤ q −2 min(ai−a ) d(e, y), we have k y ∈ B H (q −2 min(ai−a ) q −κi ). Thus, the support of all functions of the form h → f (kh) is contained in B.
If we define f h (y) = f (k y h)χ(y) = f (g −1 ai−a yg ai−a h)χ(y), then we have We apply Proposition 3.3 with q −κi in place of r, f h in place of f and g ai−a hx in place of x. Note that the injectivity radius of X is at least c 0 d for some c 0 > 0 (see the argument of Proposition 3.5 in [7]). We choose = ( 2 c0 ) 1/d q −κi/d so that for h ∈ V 2 , the projection map π = π g ai−a hx : G → X given by π g ai−a hx (g) = gg ai−a hx is injective for g ∈ B G (2q −κi ). Thus, all the assumptions of Proposition 3.3 holds and we have for δ 0 in Proposition 3.2. Since the mapping η : y → k y is contracting, the 2 -norm of the l-th level projection of f will not increase after precomposition with η for each l ≥ 0. It follows from Lemma 3.1 that Note that S d 2 (χ) q (m 2 +mn+n 2 )κi and E(q −κi , φ) q (mn+1)κi . Recall that −1 ≤ i 2(m+n) < and hence, if κ > 0 is small enough, then δ 0 min a/ log q > κi(d 2 + 1). This completes the proof.
Let us define The following double equidistribution is the main ingredient of the proof of Theorem 1.1.

Proposition 3.5 (Double equidistribution).
There exists δ 2 > 0 with the following property. Given f ∈ C ∞ c (H), φ, ψ ∈ C ∞ c (X) and a compact subset L of X, there exists C 2 = C 2 (f, φ, ψ, L) such that for any x 1 , x 2 ∈ L and i, j ∈ N with i ≤ j, we have Proof. It is enough to show the statement for sufficiently large i, j and continuous characteristic functions f, φ and ψ. Choose compact open subgroups H f of H and K φ of G which leave f and φ invariant, respectively. By taking l large enough, we may assume that the l-th congruence subgroup of K satisfies K (l) ∩ H ⊂ H f and K (l) ⊂ K φ . Let q ∈ C ∞ c (H) be such that q ≥ 0, H q(y)dy = 1 and supp (q) ⊂ K (l) ∩ H. Similar to the proof of Theorem 3.4, choose i 0 large enought so that Making a change of variable h → k y h gives us where we used K φ -invariance of φ and the fact that y ∈ supp (q) Hence by applying Theorem 3.4 with g i a hx 2 in place of x, and X in place of L. Finally, Therefore, for some r ∈ N, we get with an implied constant depending only on the degree-r Sobolev norm of f ,φ and ψ. This completes the proof of the Proposition.
Proof. The proof is verbatim the proof of Lemma 3.2 in [8]. The following inequalities give the conclusion.
In particular, let L s be the set of intervals of the form [2 k l, 2 k (l + 1)) ∩ Z ≥0 where 2 k (l + 1) < 2 s . Then

SANGHOON KWON AND SEONHEE LIM
For any given > 0, let for y / ∈ Y s and for all 1 ≤ r ≤ 2 s . Borel-Cantelli lemma implies the following corollary.
Corollary 3.7. With the notations as above, we have a measurable subset Z of Y with full measure such that for every y ∈ Z there exists s y ∈ N such that y / ∈ Y s whenever s ≥ s y .
Proof. The proof is similar with Lemma 3.4 and Lemma 3.5 in [8]. Given x ∈ X, φ ∈ C c (X) and > 0, let ν be the probability measure on X defined by Let α = X ϕdµ. Then there exist C > 0 and δ > 0 such that holds for every m ≥ n ≥ 0. Applying Corollary 3.7 to F (x, n) = φ(g n a x) − α, we obtain a full measure subset Z of Y with the following property. For every y ∈ Z , we have 4. Diophantine approximation with weights. As mentioned in the introduction, a quantitative Khintchine-Groshev type theorem over a field of formal series is obtained in [3] as follows. Let V = {q −n : n ∈ N} and φ : R + → V . For p ∈ K n and q ∈ K m , consider the inequalities Let > 0 be arbitrary and let Φ(T ) = (q m − 1) T r=0 q rm φ(q r ) n .

EQUIDISTRIBUTION IN A LOCAL FIELD OF POSITIVE CHARACTERISTIC 181
The number of solutions (p, q) ∈ Z m × Z n satisfying (4) is for λ mn -almost every A ∈ M . We consider the weighted quasi-norms Define the set Note that Remark that this term is zero if ka i are not integers for all i. Similarly, Thus λ m+n (E T,R ) is equal to k∈Q,0≤k≤T The above sum is in fact a sum over rational numbers of the form l/a j for some integer l and for some m+1 ≤ j ≤ m+n, thus it is a finite sum and it is well-defined.
We further remark that E t,R − E t−1,R = E 1,R for any t ∈ N, thus it also satisfies the homogeneity property with respect to positive integers There is a one-to-one correspondence between the set of nonzero solutions of and the intersection of u A Γ with the set E T,R . Modifying the argument of [3], we can compute the number N R (T, A) of solutions satisfying (5). for λ mn -almost every A ∈ Mat m×n (K).

SANGHOON KWON AND SEONHEE LIM
Proof. Given a vector q ∈ K m , let us define Modifying the proof of [3] Proposition 3, for q β = q k we have Moreover, by Proposition 4 of loc. cit. it holds that for linearly independent vectors q, q ∈ Z n . Let d(q) be the number of common divisors in Z of the coordinates of q and let for q β = q k . Since only the elements corresponding to the pairs of parallel vectors q and q contribute, we have for s < t. The Lemma 10 in ( [19]) says that given a measure space (Ξ, ω), if a sequence of nonnegative ω-measurable function {f k } and two sequences of nonnegative real numbers {a k } and {b k } satisfy 0 ≤ a k ≤ b k ≤ 1 and for every pair of integers (i, j) with i < j, then for ω-almost every x where B(n) = 1≤k≤n b k . It follows that we have for λ mn -almost every A, for S(T ) = 1≤ q β ≤q T τ q . We immediately obtain It remains to show that S(T ) = O(Ψ R (T ) log Ψ R (T )). In fact, using the Dirichlet series of the number of monic divisors ( [14], page 17), we have This completes the proof.
Switching the roles of x and y, we define Since both of these sets have measure q R(a1+···+am)−1 , we get the conclusion.
Proof. It holds that for all lattices Λ. Now together with Theorem 4.1 implies the first equation. For the second equality, if we Therefore, lim N →∞ holds for all ϕ ∈ C c (X) and A ∈ M . For any > 0, there exists Since was arbitrary, for almost every A ∈ M we have which completes the proof.

5.
Equidistribution with respect to C α (X) and counting with directions. For a generalized topological space E equipped with a measure λ on which a unimodular group G acts transitively and preserving λ, the mean value theorem of Siegel [18] has been generalized in [12]. In our special case when E = K d , G = SL(d, K) and Γ = SL(d, Z), we have In order to control the rate of growth at infinity of unbounded functions on X, let us use the following notation introduced by [5]: For a unimodular lattice Λ ∈ X and a subgroup ∆ ≤ Λ with L = K∆, let us denote by d(∆) the volume of L/∆ and α(Λ) = max{d(∆) −1 : ∆ ≤ Λ}. Proof. Since χ B q r and α is invariant under the left action of the group SL(d, O), we may assume that Λ = t a1 Ze 1 + · · · + t a d Ze d with a 1 ≤ a 2 ≤ · · · ≤ a k ≤ 0 < a k+1 ≤ · · · ≤ a j ≤ r < a j+1 ≤ · · · ≤ a d and d i=1 a i = 0.
This completes the proof.