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Sets of integers with large trigonometric sums. (English) Zbl 0958.11055
Deshouillers, Jean-Marc (ed.) et al., Structure theory of set addition. Paris: Société Mathématique de France, Astérisque. 258, 35-76 (1999).
Let $$k$$ be a positive integer and $$u< k$$ a positive real. For a set $$K= \{a_1< a_2<\cdots< a_k\}$$, $$a_j\in \mathbb{Z}$$, $$1\leq j\leq k$$, let $$S_K(\alpha)= \sum_{j=1}^k e^{2\pi i\alpha a_j}$$, $$s_K(\alpha)= |S_K(\alpha)|$$, $$E_{K,u}= \{\alpha\in [0,1): s_K(\alpha)\geq k-u\}$$ and $$\mu_K(u)= \mu(E_{K,u})$$, where $$\mu$$ is the Lebesgue measure on $$[0,1]$$ normalized so that $$\mu ([0,1])= 1$$. Denote by $$\mu_{\max} (k,u)$$ the supremum of $$\mu_K(u)$$ on all sets $$K$$ of size $$k$$. Freiman and Yudin studied the problem of finding $$\mu_{\max} (k,u)$$. Freiman proved that if $$u=1$$, $$a_1=0$$ and $$a_k< 0,05 k^{3/2}$$, the maximal measure is $\mu_{\max} (k,u)= \frac{2\sqrt{6}}{\pi} k^{-3/2}+ O(k^{-2})$ and it is attained by $$K$$ if and only if $$K$$ is an arithmetic progression, and Yudin proved that if $$u= o(k)$$ then $\mu_{\max} (k,u)= \frac{2\sqrt{6}}{\pi} \frac 1k \biggl( \frac uk\biggr)^{1/2} (1+o(1))\quad \text{as }k\to \infty.$ In this paper the author estimates $$\mu(E_{K,u})$$ in the case where $$K$$ is an arithmetic progression and proves the lower bound for such a progression: $\mu(E_{K,u})\geq \frac{2\sqrt{6}}{\pi} \frac 1k \biggl( \frac uk\biggr)^{1/2}.$ He also proves that the upper bound $$\mu(E_{K,u})\leq \frac dk (\frac uk)^{1/2}$$ holds for an explicitly given $$d\approx 4$$ and all sets $$K$$ under some mild restrictions on $$k$$ and $$u$$.
Some other results of a similar nature are also obtained.
For the entire collection see [Zbl 0919.00044].

##### MSC:
 11L03 Trigonometric and exponential sums, general 42A05 Trigonometric polynomials, inequalities, extremal problems