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- Author or Editor: Artūras Dubickas x

Let
*α*
be an algebraic number which is not a root of a rational number. We show that the discriminant of
*α*
^{n}
tends to infinity with
*n*
tending to infinity and give a lower bound for this discriminant in terms of the degree of
*α*
, its Mahler’s measure and
*n*
.

## 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.