(k+1)-sums versus k-sums

Details (k+1)-sums versus k-sums (k+1)-sums versus k-sums

2010-11-19

arxiv.org/abs/1011.4495v1

Mathematics

Number Theory

5 pages

Scientific paper

Given $k\in \mathbb{N}$ and a subset $A=\{a_{1},...,a_{n}\}$ of $n$ integers, we define $S_{k}=S_{k}(A)=\{\sum_{I}a_{i}:I\in [n]^{(k)}\}$, i.e. $S_{k}$ is the set of integers that may be expressed as a sum of $k$ elements of $A$ (with repetitions not allowed). We prove that if $n\ge (k^{2}+7k)/2$ then the ratio $|S_{k+1}|/|S_{k}|$ is maximised by taking $A$ to be a geometric progression. This answers a question of Ruzsa.

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