Sets of integers with large trigonometric sums

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Abstract

We investigate the problem of optimizing, for a fixed integer k and real u and on all sets K = {a1 < a2 < ⋯ < ak} ⊂ ℤ, the measure of the set of α ∈ [0, 1] where the absolute value of the trigonometric sum SK(α) = ∑j=1k e2πiαaj is greater than k - u. When u is sufficiently small with respect to k we are able to construct a set Kex which is very close to optimal. This set is a union of a finite number of arithmetic progressions. We are able to show that any more optimal set, if one exists, has a similar structure to that of Kex. We also get tight upper and lower bounds on the maximal measure.

Original languageEnglish
Pages (from-to)35-76
Number of pages42
JournalAsterisque
Volume258
StatePublished - 1 Dec 1999
Externally publishedYes

Keywords

  • Trigonometric sums

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