Let \(G\) be an abelian group, to be thought of as discrete. For a finite symmetric subset \(A \subseteq G\), one can ask how large the negative Fourier coefficients of the indicator function \(1_A\) can be. (Note that the largest positive Fourier coefficient is trivially equal to the size of the set \(A\). Also, since the set \(A\) is symmetric, the Fourier coefficients of \(1_A\) are real, thus the question regarding the maximal negative value of the Fourier transform is well defined.)
We shall give a very brief history of the problem before stating the results of this paper. For a set \(A\subseteq G\) as above, define
\[ M_G(A)=\sup_{\gamma \in \widehat{G}} -\widehat{1_A}(\gamma). \]
S. Chowla [J. Reine Angew. Math. 217, 128–132 (1965; Zbl 0127.02104)] asked for a lower bound on \(M_{\mathbb Z}(A)\). A simple averaging argument and the Littlewood conjecture [S. V. Konyagin, Izv. Akad. Nauk SSSR, Ser. Mat. 45, 243–265 (1981; Zbl 0493.42004); O. C. McGehee, L. Pigno and B. Smith, Ann. Math. (2) 113, 613–618 (1981; Zbl 0473.42001)] imply that \(M_{\mathbb Z}(A) =\Omega(\log|A|)\). The best known bound is due to I. Z. Ruzsa [Acta Arith. 111, No. 2, 179–186 (2004; Zbl 1154.11312)] and of the form \(M_{\mathbb Z}(A) =\exp(\Omega(\sqrt{\log|A|}))\).
Littlewood’s conjecture has recently been extended to abelian groups other than \(\mathbb Z\) by B. Green and S. Konyagin [Can. J. Math. 61, No. 1, 141-164 (2009; Zbl 1232.11013)]. Their results imply, for example, that \(M_{\mathbb Z/p\mathbb Z}(A) =\log^{\Omega(1)}|A|\) for \(p\) a prime, provided that \(|A|=(p+1)/2\).
In the current paper the author is able to improve on this and obtain the bound \(M_{\mathbb Z/p\mathbb Z}(A) =\Omega(p^{1/3})\), again provided that \(|A|=(p+1)/2\). For comparison, J. Spencer showed in [Trans. Am. Math. Soc. 289, 679–706 (1985; Zbl 0577.05018)] that there exist sets \(A\subseteq\mathbb Z/p\mathbb Z\) of size \((p+1)/2\) such that \(M_{\mathbb Z/p\mathbb Z}(A) =O(p^{1/2})\).
In more general abelian groups there is a simple but devastating obstacle to the obvious extension of the above result: if \(H\) is a finite subgroup of \(G\), then \(M_G(H)=0\). The author therefore proves the following refinement, which is easily seen to imply the statement for \(\mathbb Z/p\mathbb Z\) above.
Theorem. Suppose that \(G\) is a finite abelian group and \(A\) a symmetric subset of \(G\) with \(|A|=\Omega(|G|)\). Then there is a subgroup \(H \leq G\) such that
\[ M_G(A)=|A \Delta H|^{\Omega(1)}. \]
The example of a set \(A\) consisting of a large finite subgroup together with a handful of other points shows that this result is best possible up to a power.
Finally, in order to remove the hypothesis on the density of \(A\) in the theorem above, the author allows unions of subgroups to enter the picture, but we shall not state the full result here.
The paper, and in particular the introduction, is beautifully written. It draws on a number of techniques from [B. Green and T. Sanders, Ann. Math. (2) 168, No. 3, 1025–1054 (2008; Zbl 1170.43003)], including approximately 0,1-valued functions and so-called Bourgain systems, and employs an iterative method of proof.