Glasscock, DanielMoreira, JoelRichter, Florian KarlRichter, Florian K.2021-11-262021-11-262021-11-262023-06-1410.1007/s00493-023-00008-9https://infoscience.epfl.ch/handle/20.500.14299/183232WOS:0010104304000012107.10605We 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 $\times r$- and $\times s$-invariant subsets of $[0,1]$, respectively, then $\dim_\text{H} (X+Y) = \min ( 1, \dim_\text{H} X + \dim_\text{H} Y)$. Our main result yields information on the size of the sumset $\lambda X + \eta Y$ 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.sums of cantor setshausdorff dimension of sumsetsdiscrete marstrand theoremsubtree regularityfurstenberg's sumset conjectureprojectionstext::journal::journal article::research article