Sumsets contained in infinite sets of integers

Infinite Subsets of Random Sets of Integers

There is an infinite subset of a Martin-Löf random set of integers that does not compute any Martin-Löf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a Martin-Löf random set of integers, and apply a result of Miller.

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

Consecutive Integers in High-multiplicity Sumsets

Sharpening (a particular case of) a result of Szemerédi and Vu [SV06] and extending earlier results of Sárközy [S89] and ourselves [L97b], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands. A corollary of our main result is...

Partitions of positive integers into sets without infinite progressions

We prove a result which implies that, for any real numbers a and b satisfying 0 ≤ a ≤ b ≤ 1, there exists an infinite sequence of positive integers A with lower density a and upper density b such that the sets A and N \ A contain no infinite arithmetic and geometric progressions. Furthermore, for any m ≥ 2 and any positive numbers a1, . . . , am satisfying a1 + · · · + am = 1, we give an explic...