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• W4380925156 abstract "Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a theory of locality in sumsets, with several applications to John-type approximation and stability of sets with small doubling. One highlight shows that if $A sub mb{Z}$ with $|A+A| le (1-eps) 2^d |A|$ is non-degenerate then $A$ is covered by $O(2^d)$ translates of a $d$-dimensional generalised arithmetic progression ($d$-GAP) $P$ with $|P| le O_{d,eps}(|A|)$; thus we obtain one of the polynomial bounds required by PFR, under the non-degeneracy assumption that $A$ is not efficiently covered by $O_{d,eps}(1)$ translates of a $(d-1)$-GAP. We also prove a stability result showing for any $eps,aA>0$ that if $A sub mb{Z}$ with $|A+A| le (2-eps)2^d|A|$ is non-degenerate then some $A' sub A$ with $|A'|>(1-aA)|A|$ is efficiently covered by either a $(d+1)$-GAP or $O_{aA}(1)$ translates of a $d$-GAP. This `dimension-free' bound for approximate covering makes for a surprising contrast with exact covering, where the required number of translates not only grows with $d$, but does so exponentially. Another highlight shows that if $A sub mb{Z}$ is non-degenerate with $|A+A| le (2^d + ell)|A|$ and $ell le 0.1 cdot 2^d$ then $A$ is covered by $ell+1$ translates of a $d$-GAP $P$ with $|P| le O_d(|A|)$; this is tight, in that $ell+1$ cannot be replaced by any smaller number. The above results also hold for $A sub mb{R}^d$, replacing GAPs by a suitable common generalisation of GAPs and convex bodies, which we call generalised convex progressions. In this setting the non-degeneracy condition holds automatically, so we obtain essentially optimal bounds with no additional assumption on $A$. Here we show that if $Asubsetmathbb{R}^k$ satisfies $|frac{A+A}{2}|leq (1+delta)|A|$ with $deltain(0,1)$, then $exists A'subset A$ with $|A'|geq (1-delta)|A|$ so that $|co(A')|leq O_{k,1-delta}(|A|)$. This is a dimensionally independent sharp stability result for the Brunn-Minkowski inequality for equal sets, which hints towards a possible analogue for the Pr'ekopa-Leindler inequality. These results are all deduced from a unifying theory, in which we introduce a new intrinsic structural approximation of any set, which we call the `additive hull', and develop its theory via a refinement of Freiman's theorem with additional separation properties. A further application that will be published separately is a proof of Ruzsa's Discrete Brunn-Minkowski Conjecture cite{Ruzsaconjecture}." @default.
• W4380925156 created "2023-06-17" @default.
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• W4380925156 date "2023-01-01" @default.
• W4380925156 modified "2023-10-18" @default.
• W4380925156 title "Locality in sumsets" @default.
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• W4380925156 doi "https://doi.org/10.5817/cz.muni.eurocomb23-079" @default.
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