Abstract
For set \(A\subset \mathbb{F}_p^*\) define by \(\mathsf{sf}(A)\) the size of the largest sum-free subset of A. Alon and Kleitman [3] showed that \(\mathsf{sf} (A) \ge |A|/3+O(|A|/p)\). We prove that if \(\mathsf{sf}(A)-|A|/3\) is small then the set A must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies on irregularity of distribution of multiplicative subgroups on certain intervals in \(\mathbb{F}p\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Alon, Paul Erdős and probabilistic reasoning, in: Erdős centennial, Bolyai Soc. Math. Stud., vol. 25, János Bolyai Math. Soc. (Budapest, 2013), pp. 11–33
Alon, N., Bourgain, J.: Additive patterns in multiplicative subgroups. GAFA 24, 721–739 (2014)
N. Alon and D. J. Kleitman, Sum-free subsets, in: A Tribute to Paul Erdős, Cambridge University Press (1990), pp. 13-26
Bourgain, J.: Estimates related to sumfree subsets of sets of integers. Israel J. Math. 97, 71–92 (1997)
Chang, M.-C.: A polynomial bound in Freiman's theorem. Duke Math. J. 113, 399–419 (2002)
P. Erdős, Extremal problems in number theory, in: Proc. Sympos. Pure Math., vol. VIII, Amer. Math. Soc. (Providence, RI, 1965), pp. 181–189
Eberhard, S.: Følner sequences and sum-free sets. Bull. Lond. Math. Soc. 47, 21–28 (2015)
Eberhard, S., Green, B., Manners, F.: Sets of integers with no large sum-free subset. Ann. Math. 180, 621–652 (2014)
Goldfeld, D.M.: A simple proof of Siegel's theorem. Proc. Nat. Acad. Sci. 71, 1055–1055 (1974)
Goldfeld, D.M.: The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3, 624–663 (1976)
D. M. Goldfeld, Gauss's class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. (N.S.), 13 (1985), 23–37
Green, B.: A Szemerédi-type regularity lemma in abelian groups. GAFA 15, 340–376 (2005)
Green, B.: Some constructions in the inverse spectral theory of cyclic groups. Comb. Probab. Comp. 12, 127–138 (2003)
Gross, B.H., Zagier, D.B.: Heegner points and derivatives of L-series. Invent. Math. 84, 225–320 (1986)
A. B. Kalmynin, Long nonnegative sums of Legendre symbols, arXiv:1911.10634v1 (2019)
Kneser, M.: Abschätzungen der asymptotischen Dichte von Summenmengen. Math. Z. 58, 459–484 (1953)
M. Lewko, An improved upper bound for the sum-free subset constant, J. Integer Seq., 13 Article 10.8.3, 15 (2010)
Montgomery, H., Vaughan, R.: Multiplicative Number Theory. Cambridge University Press (Cambridge, I. Classical Theory (2007)
Pólya, G.: Über die Verteilung der quadratischen Reste und Nichtreste. Göttinger Nachrichten 21–29 (1918)
S. Wright, Quadratic Residues and Non-residues: Selected Topics, Lecture Notes in Math., vol. 2171. Springer (2016)
Acknowledgements
We would like to thank Alexander Kalmynin for very useful conversation about distribution of quadratic residues and Mateusz for linguistic correction of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the 80th birthday of Endre Szemerédi
The first named author is supported by National Science Centre, Poland grant 2019/35/B/ST1/00264.
Research supported by the grant of the Russian Government N 075-15-2019-1926.
Rights and permissions
About this article
Cite this article
Schoen, T., Shkredov, I.D. L-functions and sum-free sets. Acta Math. Hungar. 161, 427–442 (2020). https://doi.org/10.1007/s10474-020-01055-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-020-01055-0