Quantitative stability for sumsets in Rn\mathbb R^n

  • Alessio Figalli

    ETH Zürich, Switzerland
  • David Jerison

    Massachusetts Institute of Technology, Cambridge, USA


Given a measurable set ARnA\subset \mathbb R^n of positive measure, it is not difficult to show that A+A=2A|A+A|=|2A| if and only if AA is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If (A+A2A)/A(|A+A|-|2A|)/|A| is small, is AA close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between AA and its convex hull in terms of (A+A2A)/A(|A+A|-|2A|)/|A|.

Cite this article

Alessio Figalli, David Jerison, Quantitative stability for sumsets in Rn\mathbb R^n. J. Eur. Math. Soc. 17 (2015), no. 5, pp. 1079–1106

DOI 10.4171/JEMS/527