On a problem of Erdos, Nathanson and Sarkozy

Abstract: In 1988, Erdős, Nathanson and Sárközy proved that if $A$ is a set of nonnegative integers with lower asymptotic density $1/k$, where $k$ is a positive integer, then $(k+1) A$ must contain an infinite arithmetic progression with difference at most $ k^2-k$, where $(k+1) A$ is the set of all sums of $k+1$ elements of $A$. They asked if $(k+1)A$ must contain an infinite arithmetic progression with difference at most $O(k)$. In this talk, we answer this problem negatively by proving that, for every sufficiently large integer $k$, there exists a set $A$ of nonnegative integers with the lower asymptotic density $1/k$ such that $(k+1)A$ does not contain an infinite arithmetic progression with difference less than $k^{1.5}$.

Joint work with Ya-Li Li.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

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