Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-09-08T08:06:41.472Z Has data issue: false hasContentIssue false

# Small subsets with large sumset: Beyond the Cauchy–Davenport bound

Published online by Cambridge University Press:  21 February 2024

*
Corresponding author: Huy Tuan Pham; Email: huypham@stanford.edu

## Abstract

For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling $\kappa =|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega (\!\min\! (\kappa ^{1/3},s)|A|)$. Thus, a sumset significantly larger than the Cauchy–Davenport bound can be guaranteed by a bounded size subset assuming that the doubling $\kappa$ is large. Building up on the same ideas, we resolve a conjecture of Bollobás, Leader and Tiba that for subsets $A,B$ of $\mathbb{F}_p$ of size at most $\alpha p$ for an appropriate constant $\alpha \gt 0$, one only needs three elements $b_1,b_2,b_3\in B$ to guarantee $|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$. Allowing the use of larger subsets $A'$, we show that for sets $A$ of bounded doubling, one only needs a subset $A'$ with $o(|A|)$ elements to guarantee that $A+A'=A+A$. We also address another conjecture and a question raised by Bollobás, Leader and Tiba on high-dimensional analogues and sets whose sumset cannot be saturated by a bounded size subset.

Type
Paper

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

## Footnotes

Research of Jacob Fox is supported by a Packard Fellowship and by NSF Awards DMS-1953990 and DMS-2154129. Research of Sammy Luo is supported by NSF GRFP Grant DGE-1656518 and NSF Award No. 2303290. Research of Huy Tuan Pham is supported by a Two Sigma Fellowship, a Clay Research Fellowship and a Stanford Science Fellowship. Research of Yunkun Zhou is supported by NSF GRFP Grant DGE-1656518.

## References

Balog, A. and Szemerédi, E. (1994) A statistical theorem of set addition. Combinatorica 14(3) 263268.CrossRefGoogle Scholar
Behrend, F. A. (1946) On sets of integers which contain no three terms in arithmetical progression. Proc. Natl. Acad. Sci. U. S. A. 32(12) 331332.CrossRefGoogle ScholarPubMed
Bilu, Y. (1999) Structure of sets with small sumset. Astérisque 258 77108.Google Scholar
Bilu, Y., Lev, V. and Ruzsa, I. Z. (1998) Rectification principles in additive number theory. Discrete Comput. Geom. 19 343353.CrossRefGoogle Scholar
Bollobás, B., Leader, I. and Tiba, M. Large sumsets from small subsets, preprint, arXiv: 2204.07559.Google Scholar
Bollobás, B., Leader, I. and Tiba, M. Large sumsets from medium-sized subsets, preprint, arXiv: 2206.09366.Google Scholar
Cauchy, A. Recherches sur les nombres, J. École Polytechnique 9 (1813), 99-116. Also de Cauchy, A.L., Recherches sur les nombres, Œuvres (2nd series, Paris, 1882), Vol. 1, pp. 3963.Google Scholar
Davenport, H. (1935) On the addition of residue classes. J. Lond. Math. Soc. 10(1) 3032.CrossRefGoogle Scholar
Davenport, H. (1947) A historical note. J. Lond. Math. Soc. 22(2) 100101.CrossRefGoogle Scholar
Elkin, M. (2011) An improved construction of progression-free sets. Israel J. Math. 184(1) 93128.CrossRefGoogle Scholar
Ellenberg, J. (2017) Sumsets as unions of sumsets of subsets. Discrete Anal. Paper 14, 5 pp.Google Scholar
Fox, J. (2011) A new proof of the graph removal lemma. Ann. Math. 174 561579.CrossRefGoogle Scholar
Fox, J. and Lovász, L. M. (2017) A tight bound for Green’s arithmetic triangle removal lemma in vector spaces. Adv. Math. 321 287297.CrossRefGoogle Scholar
Freiman, G. A. Foundations of a structural theory of set addition (in Russian), Kazan, 1959; English Translation: Translation of Mathematical Monographs 37, Amer. Math. Soc., Providence, 1973.Google Scholar
Freiman, G. A. (1987) What is the structure of K if K + K is small?, in Number Theory, New York 1984–1985, Lecture Notes in Math. 1240. Springer, 109134.Google Scholar
Green, B. and Ruzsa, I. Z. (2007) Freiman’s theorem in an arbitrary abelian group. J. Lond. Math. Soc. 75(1) 163175.CrossRefGoogle Scholar
Green, B. and Tao, T. (2006) Compressions, convex geometry and the Freiman–Bilu theorem. Q. J. Math 57(4) 495504.CrossRefGoogle Scholar
Green, B. and Wolf, J. (2010) A note on Elkin’s improvement of Behrend’s construction, in Additive Number Theory: Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson. 1st ed. Springer-Verlag, 141144.CrossRefGoogle Scholar
Gowers, W. T. (2001) A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11(3) 465588.CrossRefGoogle Scholar