Let σ be a constant in the interval (0, 1), and let A be an infinite set of positive integers which contains at least c1xσ and at most c2xσ 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 \({{c\left( \sigma \right)\sqrt {\log x} } \mathord{\left/ {\vphantom {{c\left( \sigma \right)\sqrt {\log x} } {\log }}} \right. \kern-\nulldelimiterspace} {\log }}\) log x for x large enough. An example showing that this function cannot be greater than ɛ log x is also given. Another example shows that there is a set of positive integers A which contains at least xσ and at most xσ+ɛ elements in [1, x] such that A + A is greater than A only by a constant factor. The proof of the main result is based on an effective version of Freiman’s theorem due to Mei-Chu Chang.
['Artūras Dubickas', 'Paulius Šarka']
Sumsets of sparse sets