Sumsets of semiconvex sets

Semiconvex Hulls of Quasiconformal Sets

We make some remarks concerning the p-semiconvex hulls of the quasiconformal sets, using a recent significant observation of T. Iwaniec in the paper [7] on the important relation between the regularity of quasiregular mappings in the theory of geometric functions and the notion of Morrey’s quasiconvexity in the calculus of variations. We also point out several partial results on a conjecture in...

Sumsets of sparse sets

Let σ be a constant in the interval (0, 1), and let A be an infinite set of positive integers which contains at least c1x σ and at most c2x σ elements in the interval [1, x] for some constants c2 > c1 > 0 independent of x and each x ≥ x0. We prove that then the sumset A + A has more elements than A (counted up to x) by a factor c(σ) √ log x/ log log x for x large enough. An example showing that...

Sumsets of Sidon sets

On sumsets of dissociated sets

in groups F2 is found. Using our approach, we easily prove a recent result of J. Bourgain on sets of large exponential sums and obtain a tiny improvement of his theorem. Besides an inverse problem is considered in the article. Let Q be a set belonging to a sumset of two dissociated sets such that equation (1) has many solutions. We prove that in the case the large proportion of Q is highly stru...

Extensions of convex and semiconvex functions and intervally thin sets

We call A ⊂ RN intervally thin if for all x, y ∈ RN and ε > 0 there exist x′ ∈ B(x, ε), y′ ∈ B(y, ε) such that [x′, y′] ∩ A = ∅. Closed intervally thin sets behave like sets with measure zero (for example such a set cannot ”disconnect” an open connected set). Let us also mention that if the (N − 1)-dimensional Hausdorff measure of A is zero, then A is intervally thin. A function f is preconvex ...