Sparse generalised polynomials

Byszewski, Jakub; Konieczny, Jakub

doi:10.1090/tran/7257

Sparse generalised polynomials
HTML articles powered by AMS MathViewer

by Jakub Byszewski and Jakub Konieczny PDF

Trans. Amer. Math. Soc. 370 (2018), 8081-8109 Request permission

Abstract:

We investigate generalised polynomials (i.e., polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$.

Our main result is that the set of integers where a sparse generalised polynomial takes nonzero value cannot contain a translate of an IP set. We also study some explicit constructions and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomials. Finally, we show that any sufficiently sparse $\{0,1\}$-valued sequence is given by a generalised polynomial.

References

Shigeki Akiyama, Cubic Pisot units with finite beta expansions, Algebraic number theory and Diophantine analysis (Graz, 1998) de Gruyter, Berlin, 2000, pp. 11–26. MR 1770451
James E. Baumgartner, A short proof of Hindman’s theorem, J. Combinatorial Theory Ser. A 17 (1974), 384–386. MR 354394, DOI 10.1016/0097-3165(74)90103-4
Vitaly Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, Ultrafilters across mathematics, Contemp. Math., vol. 530, Amer. Math. Soc., Providence, RI, 2010, pp. 23–47. MR 2757532, DOI 10.1090/conm/530/10439
Jakub Byszewski and Jakub Konieczny, Automatic sequences and generalised polynomials, arXiv:1705.08979[math.NT].
Vitaly Bergelson and Alexander Leibman, Distribution of values of bounded generalized polynomials, Acta Math. 198 (2007), no. 2, 155–230. MR 2318563, DOI 10.1007/s11511-007-0015-y
V. Bergelson and R. McCutcheon, Idempotent ultrafilters, multiple weak mixing and Szemerédi’s theorem for generalized polynomials, J. Anal. Math. 111 (2010), 77–130. MR 2747062, DOI 10.1007/s11854-010-0013-4
Nicolas Chevallier, Best simultaneous Diophantine approximations and multidimensional continued fraction expansions, Mosc. J. Comb. Number Theory 3 (2013), no. 1, 3–56. MR 3284107
Nataliya Chekhova, Pascal Hubert, and Ali Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, J. Théor. Nombres Bordeaux 13 (2001), no. 2, 371–394 (French, with English and French summaries). MR 1879664
Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. of Math. (2) 164 (2006), no. 2, 513–560. MR 2247967, DOI 10.4007/annals.2006.164.513
Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, Graduate Texts in Mathematics, vol. 259, Springer-Verlag London, Ltd., London, 2011. MR 2723325, DOI 10.1007/978-0-85729-021-2
H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Analyse Math. 45 (1985), 117–168. MR 833409, DOI 10.1007/BF02792547
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), no. 2, 465–540. MR 2877065, DOI 10.4007/annals.2012.175.2.2
Inger Johanne Håland, Uniform distribution of generalized polynomials, J. Number Theory 45 (1993), no. 3, 327–366. MR 1247389, DOI 10.1006/jnth.1993.1082
Inger Johanne Håland, Uniform distribution of generalized polynomials of the product type, Acta Arith. 67 (1994), no. 1, 13–27. MR 1292518, DOI 10.4064/aa-67-1-13-27
Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
Inger Johanne Håland and Donald E. Knuth, Polynomials involving the floor function, Math. Scand. 76 (1995), no. 2, 194–200. MR 1354576, DOI 10.7146/math.scand.a-12534
P. Hubert and A. Messaoudi, Best simultaneous Diophantine approximations of Pisot numbers and Rauzy fractals, Acta Arith. 124 (2006), no. 1, 1–15. MR 2262136, DOI 10.4064/aa124-1-1
A. Ya. Khintchine, Continued fractions, P. Noordhoff Ltd., Groningen, 1963. Translated by Peter Wynn. MR 0161834
J. C. Lagarias, Best simultaneous Diophantine approximations. II. Behavior of consecutive best approximations, Pacific J. Math. 102 (1982), no. 1, 61–88. MR 682045
Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190 (French). MR 0088496
A. Leibman, Polynomial sequences in groups, J. Algebra 201 (1998), no. 1, 189–206. MR 1608723, DOI 10.1006/jabr.1997.7269
A. Leibman, Polynomial mappings of groups, Israel J. Math. 129 (2002), 29–60. MR 1910931, DOI 10.1007/BF02773152
A. Leibman, Pointwise convergence of ergodic averages for polynomial actions of ${\Bbb Z}^d$ by translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 215–225. MR 2122920, DOI 10.1017/S0143385704000227
A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), no. 1, 201–213. MR 2122919, DOI 10.1017/S0143385704000215
A. Leibman, A canonical form and the distribution of values of generalized polynomials, Israel J. Math. 188 (2012), 131–176. MR 2897727, DOI 10.1007/s11856-011-0097-2
A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951 (1951), no. 39, 33. MR 0039734
William Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771. MR 260975, DOI 10.2307/2373350
Wolfgang M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526–551. MR 314761, DOI 10.2307/1970824
Pavel Zorin-Kranich. Ergodic theorems for polynomials in nilpotent groups. PhD thesis, Universiteit van Amsterdam, September 2013.