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doi: 10.3934/dcds.2021175
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[3]

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

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

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

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

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

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