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A combinatorial proof of a sumset conjecture of Furstenberg
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Glasscock, Daniel, Moreira, Joel and Richter, Florian (2023) A combinatorial proof of a sumset conjecture of Furstenberg. Combinatorica . doi:10.1007/s00493-023-00008-9 ISSN 0209-9683. (In Press)
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Official URL: https://doi.org/10.1007/s00493-023-00008-9
Abstract
We give a new proof of a sumset conjecture of Furstenberg that was rst proved by Hochman and Shmerkin in 2012: if log r= log s is irrational and X and Y are r- and sinvariant subsets of [0; 1], respectively, then dimH(X + Y ) = min(1; dimHX + dimHY ). Our main result yields information on the size of the sumset X + Y uniformly across a compact set of parameters at xed scales. The proof is combinatorial and avoids the machinery of local entropy averages and CP-processes, relying instead on a quantitative, discrete Marstrand projection theorem and a subtree regularity theorem that may be of independent interest.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Cantor sets, Combinatorial number theory | ||||||||
| Journal or Publication Title: | Combinatorica | ||||||||
| Publisher: | Springer | ||||||||
| ISSN: | 0209-9683 | ||||||||
| Official Date: | 2023 | ||||||||
| Dates: |
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| DOI: | 10.1007/s00493-023-00008-9 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | In Press | ||||||||
| Reuse Statement (publisher, data, author rights): | The version of record of this article, first published in Combinatorica, is available online at Publisher’s website: http://dx.doi.org/10.1007/s00493-023-00008-9 | ||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Date of first compliant deposit: | 14 March 2022 | ||||||||
| Date of first compliant Open Access: | 23 June 2023 | ||||||||
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