Quantitative stability for sumsets in R[superscript n]

Author(s)
Figalli, Alessio; Jerison, David S
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Abstract
Given a measurable set A ⊂ ℝ[superscript n] of positive measure, it is not difficult to show that |A + A| = |2A| if and only if A is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If (|A + A| - |2A|)/|A| is small, is A close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between A and its convex hull in terms of (|A + A| - |2A|)/|A|. Keywords: Quantitative stability, sumsets, Freiman’s theorem
Date issued
2015-05
URI
http://hdl.handle.net/1721.1/115583
Department
Massachusetts Institute of Technology. Department of Mathematics
Journal
Journal of the European Mathematical Society
Publisher
European Mathematical Publishing House
Citation
Figalli, Alessio, and David Jerison. “Quantitative Stability for Sumsets in R[superscript n].” Journal of the European Mathematical Society, vol. 17, no. 5, 2015, pp. 1079–106.
Version: Author's final manuscript
ISSN
1435-9855
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