A combinatorial proof of a sumset conjecture of Furstenberg

We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if log r / log s is irrational and X and Y are × r - and × s -invariant subsets of [0, 1], respectively, then dim H ( X + Y ) = min ( 1 , dim H X + dim H Y ) . Our main result yields info...

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Veröffentlicht in:Combinatorica (Budapest. 1981) 2023-04, Vol.43 (2), p.299-328
Hauptverfasser: Glasscock, Daniel, Moreira, Joel, Richter, Florian K.
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Sprache:eng
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Zusammenfassung:We give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if log r / log s is irrational and X and Y are × r - and × s -invariant subsets of [0, 1], respectively, then dim H ( X + Y ) = min ( 1 , dim H X + dim H Y ) . Our main result yields information on the size of the sumset λ X + η Y uniformly across a compact set of parameters at fixed 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.
ISSN:0209-9683
1439-6912
DOI:10.1007/s00493-023-00008-9