Sumsets of dense sets and sparse sets

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...

Dense point sets have sparse Delaunay triangulations

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of n points in R3 with spread ∆ has complexity O(∆3). This bound is tight in the worst case for all ∆ = O( √ n). In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to re...

Arithmetic Progressions in Sets with Small Sumsets

We present an elementary proof that if A is a finite set of numbers, and the sumset A+G A is small, |A+G A| ≤ c|A|, along a dense graph G, then A contains k-term arithmetic progressions.