## Abstract

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

## Abstract

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}≧3*b*
_{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}=3*b*
_{1}+4, then there is no sequence *A* of positive integers for which *P*(*A*)=ℕ∖*B*.

## 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 *x* ≥ *x*_{0}. We prove that then the sumset *A + A* has more elements than *A* (counted up to *x*) by a factor

*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.

## Abstract

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.

## Abstract

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.