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- W4286980438 abstract "We prove a new class of low-energy decompositions which, amongst other consequences, imply that any finite set $A$ of integers may be written as $A = B cup C$, where $B$ and $C$ are disjoint sets satisfying [ |{ (b_1, dots, b_{2s}) in B^{2s} | b_1 + dots + b_{s} = b_{s+1} + dots + b_{2s}}| ll_{s} |B|^{2s - (log log s)^{1/2 - o(1)}} ] and [ |{ (c_1, dots, c_{2s}) in C^{2s} | c_1 dots c_{s} = c_{s+1} dots c_{2s} }| ll_{s} |C|^{2s - (log log s)^{1/2 - o(1)}}.] This generalises previous results of Bourgain--Chang on many-fold sumsets and product sets to the setting of many-fold energies, albeit with a weaker power saving, consequently confirming a speculation of Balog--Wooley. We further use our method to obtain new estimates for $s$-fold additive energies of $k$-convex sets, and these come arbitrarily close to the known lower bounds as $s$ becomes sufficiently large." @default.
- W4286980438 created "2022-07-25" @default.
- W4286980438 creator A5029142792 @default.
- W4286980438 date "2021-09-10" @default.
- W4286980438 modified "2023-09-26" @default.
- W4286980438 title "Energy estimates in sum-product and convexity problems" @default.
- W4286980438 doi "https://doi.org/10.48550/arxiv.2109.04932" @default.
- W4286980438 hasPublicationYear "2021" @default.
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