Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T02:27:46.420Z Has data issue: false hasContentIssue false

The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit

Published online by Cambridge University Press:  20 November 2018

Ernie Croot*
Affiliation:
Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, U.S.A. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

How few three-term arithmetic progressions can a subset $S\,\subseteq \,{{\mathbb{Z}}_{N}}\,:=\,\mathbb{Z}\,/\,N\mathbb{Z}$ have if $|S|\,\ge \,\upsilon N$ (that is, $S$ has density at least $\upsilon$)? Varnavides showed that this number of arithmetic progressions is at least $c(v)\,{{N}^{2}}$ for sufficiently large integers $N$. It is well known that determining good lower bounds for $c\left( \upsilon \right)\,>\,0$ is at the same level of depth as Erdös's famous conjecture about whether a subset $T$ of the naturals where $\sum{_{n\in T}\,1/n}$ diverges, has a $k$-term arithmetic progression for $k\,=\,3$ (that is, a three-term arithmetic progression).

We answer a question posed by B. Green about how this minimial number of progressions oscillates for a fixed density $\upsilon$ as $N$ runs through the primes, and as $N$ runs through the odd positive integers.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Green, B., Roth's theorem in the primes. Ann. of Math. 161(2005), no. 3, 16091636.Google Scholar
[2] Green, B. and Ruzsa, I., Counting sumsets and sum-free sets modulo a prime. Studia Sci. Math. Hungar. 41(2004), no. 3, 285293.Google Scholar
[3] Varnavides, P., On certain sets of positive density. J. London Math. Soc. 34(1959), 358360.Google Scholar