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Quantitative stability for sumsets in ℝ n

Alessio Figalli, David Jerison (2015)

Journal of the European Mathematical Society

Given a measurable set A ⊂ ℝ n of positive measure, it is not difficult to show that | A + A | = | 2 A | 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 | - | 2 A | ) / | 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 | - | 2 A | ) / | A | .

Quasilinear waves and trapping: Kerr-de Sitter space

Peter Hintz, András Vasy (2014)

Journées Équations aux dérivées partielles

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally...

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