April  2022, 42(4): 1855-1871. doi: 10.3934/dcds.2021175

Sublacunary sets and interpolation sets for nilsequences

Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, OH 43210, USA

Received  August 2021 Revised  October 2021 Published  April 2022 Early access  November 2021

A set $ E \subset \mathbb{N} $ is an interpolation set for nilsequences if every bounded function on $ E $ can be extended to a nilsequence on $ \mathbb{N} $. Following a theorem of Strzelecki, every lacunary set is an interpolation set for nilsequences. We show that sublacunary sets are not interpolation sets for nilsequences. Here $ \{r_n: n \in \mathbb{N}\} \subset \mathbb{N} $ with $ r_1 < r_2 < \ldots $ is sublacunary if $ \lim_{n \to \infty} (\log r_n)/n = 0 $. Furthermore, we prove that the union of an interpolation set for nilsequences and a finite set is an interpolation set for nilsequences. Lastly, we provide a new class of interpolation sets for Bohr almost periodic sequences, and as a result, obtain a new example of interpolation set for $ 2 $-step nilsequences which is not an interpolation set for Bohr almost periodic sequences.

Citation: Anh N. Le. Sublacunary sets and interpolation sets for nilsequences. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1855-1871. doi: 10.3934/dcds.2021175
References:
[1]

L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53. Princeton University Press, Princeton, N.J., 1963.

[2]

V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences, Invent. Math., 160 (2005), 261–303. With an appendix by I. Ruzsa. doi: 10.1007/s00222-004-0428-6.

[3]

V. Bergelson and A. Leibman, IPr*-recurrence and nilsystems, Adv. Math., 339 (2018), 642-656.  doi: 10.1016/j.aim.2018.09.032.

[4] L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications, Cambridge University Press, Cambridge, 1990. 
[5]

N. Frantzikinakis, Equidistribution of sparse sequences on nilmanifolds, J. Anal. Math., 109 (2009), 353-395.  doi: 10.1007/s11854-009-0035-y.

[6]

N. Frantzikinakis, Some open problems on multiple ergodic averages, Bull. Hellenic Math. Soc., 60 (2016), 41-90. 

[7]

N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., 30 (2017), 67-157.  doi: 10.1090/jams/857.

[8]

N. Frantzikinakis and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math., 187 (2018), 869-931.  doi: 10.4007/annals.2018.187.3.6.

[9]

H. Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 211-234. 

[10]

C. Graham and K. Hare, Interpolation and Sidon Sets for Compact Groups, CMS Books in Mathematics. Boston, MA: Springer US, 2013.

[11]

B. Green and T. Tao, Linear equations in primes, Ann. of Math., 171 (2010), 1753-1850.  doi: 10.4007/annals.2010.171.1753.

[12]

B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math., 175 (2012), 541-566.  doi: 10.4007/annals.2012.175.2.3.

[13]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.

[14]

B. GreenT. Tao and T. Ziegler, An inverse theorem for the Gowers Us+1[N]-norm, Ann. of Math., 176 (2012), 1231-1372.  doi: 10.4007/annals.2012.176.2.11.

[15]

J. Griesmer, Special cases and equivalent forms of Katznelson's problem on recurrence, arXiv: 2108.02190.

[16]

D. Grow, A class of I0-sets, Collog. Math., 53 (1987), 111-124.  doi: 10.4064/cm-53-1-111-124.

[17]

S. Hartman, On interpolation by almost periodic functions, Colloq. Math., 8 (1961), 99-101.  doi: 10.4064/cm-8-1-99-101.

[18]

S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, Colloq. Math., 12 (1964), 23-39.  doi: 10.4064/cm-12-1-23-39.

[19]

B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, volume 236 of Mathematical Surveys and Monographs, American Mathematical Society, 2018. doi: 10.1090/surv/236.

[20]

B. HostB. Kra and A. Maass, Variations on topological recurrence, Monatsh. Math., 179 (2016), 57-89.  doi: 10.1007/s00605-015-0765-0.

[21]

K. Kunen and W. Rudin, Lacunarity and the Bohr topology, Math. Proc. Cambridge Philos. Soc., 126 (1999), 117-137.  doi: 10.1017/S030500419800317X.

[22]

A. N. Le, Interpolation sets and nilsequences, Colloq. Math., 162 (2020), 181-199.  doi: 10.4064/cm7937-9-2019.

[23]

A. N. Le, Nilsequences and multiple correlations along subsequences, Ergodic Theory Dynam. Systems, 40 (2020), 1634-1654.  doi: 10.1017/etds.2018.110.

[24]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-113.  doi: 10.1017/S0143385704000215.

[25]

A. Leibman, Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.  doi: 10.1017/etds.2013.36.

[26]

A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32. 

[27]

J.-F. Méla, Approximation diophantienne et ensembles lacunaires, Bull. Soc. Math. France Mem., 19 (1969), 26-54.  doi: 10.24033/msmf.21.

[28]

C. Ryll-Nardzewski, Concerning almost periodic extensions of functions, Colloq. Math., 12 (1964), 235-237.  doi: 10.4064/cm-12-2-235-237.

[29]

P. Sarnak, Mobius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97. 

[30]

E. Strzelecki, On a problem of interpolation by periodic and almost periodic functions, Colloq. Math., 11 (1963), 91-99.  doi: 10.4064/cm-11-1-91-99.

[31]

T. Tao and J. Teräväinen, The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures, Duke Math. J., 168 (2019), 1977-2027.  doi: 10.1215/00127094-2019-0002.

show all references

References:
[1]

L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53. Princeton University Press, Princeton, N.J., 1963.

[2]

V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences, Invent. Math., 160 (2005), 261–303. With an appendix by I. Ruzsa. doi: 10.1007/s00222-004-0428-6.

[3]

V. Bergelson and A. Leibman, IPr*-recurrence and nilsystems, Adv. Math., 339 (2018), 642-656.  doi: 10.1016/j.aim.2018.09.032.

[4] L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications, Cambridge University Press, Cambridge, 1990. 
[5]

N. Frantzikinakis, Equidistribution of sparse sequences on nilmanifolds, J. Anal. Math., 109 (2009), 353-395.  doi: 10.1007/s11854-009-0035-y.

[6]

N. Frantzikinakis, Some open problems on multiple ergodic averages, Bull. Hellenic Math. Soc., 60 (2016), 41-90. 

[7]

N. Frantzikinakis and B. Host, Higher order Fourier analysis of multiplicative functions and applications, J. Amer. Math. Soc., 30 (2017), 67-157.  doi: 10.1090/jams/857.

[8]

N. Frantzikinakis and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Ann. of Math., 187 (2018), 869-931.  doi: 10.4007/annals.2018.187.3.6.

[9]

H. Furstenberg, Poincaré recurrence and number theory, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 211-234. 

[10]

C. Graham and K. Hare, Interpolation and Sidon Sets for Compact Groups, CMS Books in Mathematics. Boston, MA: Springer US, 2013.

[11]

B. Green and T. Tao, Linear equations in primes, Ann. of Math., 171 (2010), 1753-1850.  doi: 10.4007/annals.2010.171.1753.

[12]

B. Green and T. Tao, The Möbius function is strongly orthogonal to nilsequences, Ann. of Math., 175 (2012), 541-566.  doi: 10.4007/annals.2012.175.2.3.

[13]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540.  doi: 10.4007/annals.2012.175.2.2.

[14]

B. GreenT. Tao and T. Ziegler, An inverse theorem for the Gowers Us+1[N]-norm, Ann. of Math., 176 (2012), 1231-1372.  doi: 10.4007/annals.2012.176.2.11.

[15]

J. Griesmer, Special cases and equivalent forms of Katznelson's problem on recurrence, arXiv: 2108.02190.

[16]

D. Grow, A class of I0-sets, Collog. Math., 53 (1987), 111-124.  doi: 10.4064/cm-53-1-111-124.

[17]

S. Hartman, On interpolation by almost periodic functions, Colloq. Math., 8 (1961), 99-101.  doi: 10.4064/cm-8-1-99-101.

[18]

S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions, Colloq. Math., 12 (1964), 23-39.  doi: 10.4064/cm-12-1-23-39.

[19]

B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, volume 236 of Mathematical Surveys and Monographs, American Mathematical Society, 2018. doi: 10.1090/surv/236.

[20]

B. HostB. Kra and A. Maass, Variations on topological recurrence, Monatsh. Math., 179 (2016), 57-89.  doi: 10.1007/s00605-015-0765-0.

[21]

K. Kunen and W. Rudin, Lacunarity and the Bohr topology, Math. Proc. Cambridge Philos. Soc., 126 (1999), 117-137.  doi: 10.1017/S030500419800317X.

[22]

A. N. Le, Interpolation sets and nilsequences, Colloq. Math., 162 (2020), 181-199.  doi: 10.4064/cm7937-9-2019.

[23]

A. N. Le, Nilsequences and multiple correlations along subsequences, Ergodic Theory Dynam. Systems, 40 (2020), 1634-1654.  doi: 10.1017/etds.2018.110.

[24]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-113.  doi: 10.1017/S0143385704000215.

[25]

A. Leibman, Nilsequences, null-sequences, and multiple correlation sequences, Ergodic Theory Dynam. Systems, 35 (2015), 176-191.  doi: 10.1017/etds.2013.36.

[26]

A. I. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949), 9-32. 

[27]

J.-F. Méla, Approximation diophantienne et ensembles lacunaires, Bull. Soc. Math. France Mem., 19 (1969), 26-54.  doi: 10.24033/msmf.21.

[28]

C. Ryll-Nardzewski, Concerning almost periodic extensions of functions, Colloq. Math., 12 (1964), 235-237.  doi: 10.4064/cm-12-2-235-237.

[29]

P. Sarnak, Mobius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97. 

[30]

E. Strzelecki, On a problem of interpolation by periodic and almost periodic functions, Colloq. Math., 11 (1963), 91-99.  doi: 10.4064/cm-11-1-91-99.

[31]

T. Tao and J. Teräväinen, The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures, Duke Math. J., 168 (2019), 1977-2027.  doi: 10.1215/00127094-2019-0002.

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