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Authors: Artūras Dubickas and Paulius Šarka

Abstract  

Let σ be a constant in the interval (0, 1), and let A be an infinite set of positive integers which contains at least c 1 x σ and at most c 2 x σ elements in the interval [1, x] for some constants c 2 > c 1 > 0 independent of x and each xx 0. We prove that then the sumset A + A has more elements than A (counted up to x) by a factor

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $${{c\left( \sigma \right)\sqrt {\log x} } \mathord{\left/ {\vphantom {{c\left( \sigma \right)\sqrt {\log x} } {\log }}} \right. \kern-\nulldelimiterspace} {\log }}$$ \end{document}
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 x σ and at most x σ+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.

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