A combinatorial proof of a sumset conjecture of Furstenberg

Abstract

AbstractWe give a new proof of a sumset conjecture of Furstenberg that was first proved by Hochman and Shmerkin in 2012: if $$\log r/\log s$$ log r / log s is irrational and X and Y are $$\times r$$ × r - and $$\times s$$ × s -invariant subsets of [0, 1], respectively, then $$\dim _{\text {H}}(X + Y ) = \min (1, \dim _{\text {H}}X + \dim _{\text {H}}Y )$$ dim H ( X + Y ) = min ( 1 , dim H X + dim H Y ) . Our main result yields information on the size of the sumset $$\lambda X + \eta Y$$ 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.