On convexity and sumsets

Abstract: A finite set $A$ of $n$ reals is convex if the sequence of neighbouring differences is strictly monotone. Erdös suggested that that the set of squares of the first $n$ integers may constitute the extremal case, namely that for any convex $A$, $|A+A| > n^{2-o(1)}$. The question is still open, we'll review some partial progress.

What about $k- $fold sums $A+A+A+...$? In the case of squares, they stop growing after $k=2$, and for $k$th powers they grow up to $n^k$. In a joint work with Brandon Hanson and Olly Roche-Newton we show, using elementary methods, that if $A=f([n])$, where $f$ is a real function with $k-1$ strictly monotone derivatives, taking sufficiently many sums does lead to growth up to $n^{k-o(1)}$. We generalise this by replacing the interval $[n]$ versus $f([n])$ by any set with small additive doubling versus its image by $f$, which enables us to apply this to sum-product type questions.

number theory

Audience: researchers in the topic

( video )

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Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

We will email out the link to all registered participants the day before.

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