Uniformity in the dimension of sumsets of $p$- and $q$-invariant sets, with applications in the integers

Abstract: Harry Furstenberg made a number of conjectures in the 60's and 70Õs seeking to make precise the heuristic that there is no common structure between digit expansions of real numbers in different bases. Recent solutions to his conjectures concerning the dimension of sumsets and intersections of times $p$- and $q$-invariant sets now shed new light on old problems. In this talk, I will explain how to use tools from fractal geometry and uniform distribution to get uniform estimates on the Hausdorff dimension of sumsets of times $p$- and $q$-invariant sets. I will explain how these uniform estimates lead to applications in the integers: the dimension of a sumset of a p-invariant set and a q-invariant set in the integers is as large as it can be.

This talk is based on joint work with Joel Moreira and Florian Richter.

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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