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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 $times r$- 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.
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of
We prove a conjecture of Ohba which says that every graph $G$ on at most $2chi(G)+1$ vertices satisfies $chi_ell(G)=chi(G)$.
In the context of the (generalized) Delta Conjecture and its compositional form, DAdderio, Iraci, and Wyngaerd recently stated a conjecture relating two symmetric function operators, $D_k$ and $Theta_k$. We prove this Theta Operator Conjecture, findi
A computer search through the oriented matroid programs with dimension 5 and 10 facets shows that the maximum strictly monotone diameter is 5. Thus $Delta_{sm}(5,10)=5$. This enumeration is analogous to that of Bremner and Schewe for the non-monotone
We present an infinite family of Borwein type $+ - - $ conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.