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Authors: J. Deshouillers and A. Plagne

Abstract  

We construct a Sidon set which is an asymptotic additive basis of order at most 7.

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For a set A, let P(A) be the set of all finite subset sums of A. We prove that if a sequence B={b 1<b 2<⋯} of integers satisfies b 1≧11 and b n+1≧3b n+5 (n=1,2,…), then there exists a sequence of positive integers A={a 1<a 2<⋯} for which P(A)=ℕ∖B. On the other hand, if a sequence B={b 1<b 2<⋯} of positive integers satisfies either b 1=10 or b 2=3b 1+4, then there is no sequence A of positive integers for which P(A)=ℕ∖B.

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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 c1xσ and at most c2xσ elements in the interval [1, x] for some constants c2 > c1 > 0 independent of x and each xx0. 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 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.

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B. J. Birch [1] proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form p a q b, where p, q are given coprime integers greater than 1. Subsequently, Davenport pointed out that the exponent b can be bounded in terms of p and q. N. Hegyvári [3] gave an effective version of this bound. In this paper, we improve this bound by reducing one step.

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Sharpening (a particular case of) a result of Szemerédi and Vu [4] and extending earlier results of Sárközy [3] and ourselves [2], 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, whose length is comparable with the lengths of the set summands.

A corollary of our main result is as follows. Let k,l≥1 and n≥3 be integers, and suppose that A 1,…,A k⊆[0,l] are integer sets of size at least n, none of which is contained in an arithmetic progression with difference greater than 1. If k≥2⌈(l−1)/(n−2)⌉, then the sumset A 1+⋅⋅⋅+A k contains a block of at least k(n−1)+1 consecutive integers.

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