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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].

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