## Arithmetical progressions and the number of sums.(English)Zbl 0761.11005

Let $$r_ k(n)$$ denote the maximal number of integers that can be selected from the interval $$[1,n]$$ without including a $$k$$ term arithmetical progression and write $$\omega_ k(n)=n/r_ k(n)$$. Let $$A$$ be a finite set of integers, $$| A|=n$$ and assume that $$A$$ does not contain any $$k$$-term arithmetical progression. It is proved that $| A+B|\geq\textstyle{{1\over 2}}\omega_ k(n)^{1/4} n^{1/4}| B|^{3/4}$ for every set $$B$$, in particular $$| A+A|\geq{1\over 2}\omega_ k(n)^{1/4}n$$, $$| A- A|\geq{1\over 2}\omega_ k(n)^{1/4}n$$.
It is known that $$\omega_ 3(n)\gg(\log n)^ c$$ with a positive constant $$c$$ (Heath-Brown, Szemerédi). Applying this estimate we obtain that $| A+A|\geq n(\log n)^ c, \qquad | A-A|\geq n(\log n)^ c$ for $$n>n_ 0$$, whenever $$A$$ does not contain any 3-term arithmetical progression with a positive absolute constant $$c$$. This is an effective version of a result of G. A. Freiman [Foundations of a structural theory of set addition (Kazan 1966; Zbl 0203.353); English transl. (1973)]. The proof is completely different from Freiman’s but uses his fundamental concept of isomorphism.

### MSC:

 11B25 Arithmetic progressions 11B50 Sequences (mod $$m$$) 11B75 Other combinatorial number theory

### Citations:

Zbl 0203.353; Zbl 0271.10044
Full Text:

### References:

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