## 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.

## Additional Information

**Jakub Byszewski**- Affiliation: Department of Mathematics and Computer Science, Institute of Mathematics, Jagiellonian University, ul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland
- MR Author ID: 799547
- Email: jakub.byszewski@gmail.com
**Jakub Konieczny**- Affiliation: Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
- MR Author ID: 1178795
- Email: jakub.konieczny@gmail.com
- Received by editor(s): November 28, 2016
- Received by editor(s) in revised form: March 2, 2017
- Published electronically: June 26, 2018
- Additional Notes: This research was supported by the National Science Centre, Poland (NCN), under grant no. DEC-2012/07/E/ST1/00185.

The second author also acknowledges the generous support from the Clarendon Fund and SJC Kendrew Fund for his doctoral studies. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8081-8109 - MSC (2010): Primary 37A45, 05D10, 28D05; Secondary 37B05, 11J54, 11J70, 11J71
- DOI: https://doi.org/10.1090/tran/7257
- MathSciNet review: 3852458