Bourgain 1986 sur le minimum d une somme

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Bourgain, J., Sur le minimum d'une somme de cosinus. Acta Arith. 45 (1986), 381-389.

Bourgain (writing in French) studies a fixed trigonometric polynomial f whose Fourier coefficients take only the values 0 and 1, with f-hat(0)=0 and f-hat(k)=f-hat(-k), so that f/2 is a sum of cosines. The stated goal, achieved in the paper, is the lower bound ||f^-||_inf >= 2^{(log N)^eps} for an absolute constant eps>0, where N is the number of frequencies in the spectrum and f^- is the negative part of f; this beats the earlier logarithmic-type bounds. The method is an inductive construction (Proposition of Section 1, proved by recursion on d = 0, 1, 2, ...) showing that either ||f^-||_inf is already large or the spectrum of f contains a translate of a large multidimensional grid {0,k_1,2k_1,...,dk_1} + ... + {0,k_J,...,dk_J} with J of size about (log N)^eps and the k_j highly dissociated; the argument uses test measures, L^2/L^4 moment comparisons, and a basic lemma from Roth's paper on the same problem. Bearing on problem 510 (the minimum of a cosine sum with 0-1 coefficients): this is the paper that raises the known lower bound for the negative minimum from Roth-type logarithmic size to 2^{(log N)^eps}.

Source: https://eudml.org/doc/205983. No notice is printed on the scan's first and last pages; the EuDML record states no rights, and the publisher's article page offers the PDF as a "Free download under CC-BY license", a Creative Commons Attribution license with no version or URL named (https://www.impan.pl/get/doi/10.4064/aa-45-4-381-389, read 2026-10-02).

Bears on. #510

Results to transcribe.

• Main bound (Section 0): For f a trigonometric polynomial with Fourier coefficients in {0,1}, f-hat(0)=0, f-hat(k)=f-hat(-k), and N = |Spec f|, one has ||f^-||_inf >= 2^{(log N)^eps} for an absolute eps > 0.
• Proposition (Section 1, grid construction): There exist constants 0 < eps, delta, delta_0, delta_1, delta_2 < 1 such that for F with 0-1 coefficients decomposing as F = F' + F'' + F''' with |F'| <= |f|, ||F''||_inf < 2^{(log N)^{delta_1}}, ||F'''||_1 < 2^{-(log N)^{delta_2}} and n = |Spec F| > 2^{(log N)^{delta_0}}, either ||f^-||_inf > 2^{(log N)^eps} or there are at least n/M integers alpha (M = 2^{(log N)^delta}) whose intersection of d+1 translates of the spectrum has cardinality > n/M.
• Remark (Section 1): The author restates the second alternative as the existence of a d-dissociated sequence of integers k_1 < ... < k_J with J of order (log N)^eps such that Spec F contains a translate of the grid {0,k_1,...,dk_1} + ... + {0,k_J,...,dk_J}; this construction is contained in the proof of the Proposition, and the case of interest is F = f.