On Fourier coefficients of sets with small doubling

Let ${\mathbf{G}}$ be a finite abelian group and $\widehat{{\mathbf{G}}}$ its dual group. For any function $f:{\mathbf{G}}\to\mathbb{C}$ and $\chi\in\widehat{{\mathbf{G}}}$ define the Fourier transform of $f$ at $\chi$ by the formula

This paper considers the case when the function $f$ has a special form, namely, $f$ the characteristic function of a set $A\subseteq{\mathbf{G}}$ and we want to study the quantity

where by $\chi_{0}$ we have denoted the principal character of $\widehat{{\mathbf{G}}}$ and we use the same capital letter to denote a set $A\subseteq{\mathbf{G}}$ and its characteristic function $A:{\mathbf{G}}\to\{0,1\}$. It is well–known that ${\mathcal{M}}(A)$ is directly related to the uniform distribution properties of the set $A$ (see, e.g., [13], [28]). Moreover, we impose an additional property on the set $A$, namely, that it is a set with small doubling, i.e. we consider sets $A$ for which the ratio $|A-A|/|A|$ is small. Recall that given two sets $A,B\subseteq{\mathbf{G}}$ the sumset of $A$ and $B$ is defined as

In a similar way we define the difference sets $A-A$ and the interated sumsets, e.g., $2A-A$ is $A+A-A$. In terms of the difference sets $|A-A|:=K|A|$ it is easy to see that

and estimate (3) gives us a simple lower bound for the quantity ${\mathcal{M}}(A)$. In general, bound (3) is tight, but the situation changes dramatically depending on the density $\delta:=|A|/|{\mathbf{G}}|$ of $A$. For example, in [11, Lemma 4.1] (also, see [24] and [22, Propostion 6.1]) it was proved that

provided $\delta=\exp(-\Omega(K))$. Thus, if the density is exponentially small relative to $K$, then ${\mathcal{M}}(A)$ is close to its maximum value $|A|$. On the other hand, it is easy to see we always have $\delta\leqslant K^{-1}$ and for a random set $A\subseteq{\mathbf{G}}$ one has $\delta\gg K^{-1}$. Therefore, one of the reasonable questions here is the following. Assume that $\delta\sim K^{-d}$, where $d>1$, that is, the dependence of $\delta$ on $K$ is polynomial. What non–trivial properties do the Fourier coefficients of set $A$ have in this case? Such a problem arises naturally in connection with the famous Freiman $3k-4$ theorem in $\mathbb{F}_{p}$, see, e.g., [7], [17], [18]. In particular, in [18, Theorem 6] it was proved that if $\delta\ll K^{-3}$, then

where $\kappa_{0}>0$ is an absolute constant. Thus, a non-trivial lower bound for ${\mathcal{M}}(A)$ exists in any abelian group, and hence the Fourier coefficients of sets with small doubling have some interesting properties. Other results on the quantity ${\mathcal{M}}(A)$ and its relation to sets $A_{x}$ (see Section 2 below) were obtained in [27]. The main result of this paper (all required definitions can be found in Sections 2, 4) concerns an even broader regime $\delta\ll K^{-2}$ and gives us a new structural property of sets with small doubling and small Fourier coefficients.

Let us make a few remarks regarding Theorem 10. First of all, the bounds (8)—(10) hold for any dense subset of our set $A$, which means that $A$ has a very rigid structure (for example, see the second part of Corollary 9 below). Secondly, the dependencies on parameters in (8)—(10) have polynomial nature, which distinguishes them from the best modern results on the structure of sets with small doubling, see [22], [23]. Also, it appears that Theorem 10 does not depend on the recent breakthrough progress concerning Polynomial Freiman–Ruzsa Conjecture [8], [9]. Thirdly, the inversion of bound (7) guarantees the existence of a large intersection of $A$ and a translation of a Bohr set ${\mathcal{B}}_{*}$, see inequality (8). This is quite surprising, because usually we have the opposite picture: small Fourier coefficients help to avoid such structural objects as Bohr sets. Finally, our proof drastically differs from the arguments of [11], [24], [22], which give us inequality (4). Instead of using almost periodicity of multiple convolutions [6] and the polynomial growth of sumsets, we apply some observations from the higher energies method, see papers [25], [27]. In particular, our structural subset of $A-A$ has a different nature (it resembles some steps of the proof of [8], [9] and even older constructions of Schoen, see [26, Examples 5,6]). Let us consider the following motivating example (see, e.g., [26, Example 5]).

Roughly speaking, the same Example 2 shows that Theorem 10 is tight (also, see more rigorous Example 20 below). Indeed, it is easy to see that $\widehat{A}=\widehat{H}\cdot\widehat{\Lambda}$ and assuming that $\Lambda$ is a sufficiently small set and ${\mathcal{M}}^{2}(\Lambda)\ll|\Lambda|\ll K$, we get ${\mathcal{M}}^{2}(A)=|H|^{2}{\mathcal{M}}^{2}(\Lambda)\ll|A|^{2}/K$. Thus all conditions of Theorem 10 hold but the obvious subspace in $A-A$ is of course $H$ and we have ${\rm codim}(H)=\log(\delta^{-1}K)$, which coincides with (9) up to multiple constants.

Let ${\mathbf{G}}$ be a finite abelian group and we denote the cardinality of ${\mathbf{G}}$ by $N$. Given a set $A\subseteq{\mathbf{G}}$ and a positive integer $k$, let us put

Now we have

and for $x\in A-A$ we denote by $A_{x}$ the intersection $A\cap(A+x)$. The useful inclusion of Katz–Koester [14] is the following

A natural generalization of the last formula in (12) is the set

which is called the higher difference set (see [25]). For any two sets $A,B\subseteq{\mathbf{G}}$ the additive energy of $A$ and $B$ is defined by

If $A=B$, then we simply write $\mathsf{E}(A)$ for $\mathsf{E}(A,A)$. Also, one can define the higher energy (see [25] and another formula (19) below)

Let $\widehat{{\mathbf{G}}}$ be its dual group. For any function $f:{\mathbf{G}}\to\mathbb{C}$ and $\chi\in\widehat{{\mathbf{G}}}$ we define its Fourier transform using the formula (1). The Parseval identity is

Given a function $f$ and $\varepsilon\in(0,1]$ define the spectrum of $f$ as

We need Chang’s lemma [5].

By measure we mean any non–negative function $\mu$ on ${\mathbf{G}}$ such that $\sum_{g\in{\mathbf{G}}}\mu(g)=1$. If $f,g:{\mathbf{G}}\to\mathbb{C}$ are some functions, then

One has

Having a function $f:{\mathbf{G}}\to\mathbb{C}$ and a positive integer $k>1$, we write $f^{(k)}(x)=(f\circ f\circ\dots\circ f)(x)$, where the convolution $\circ$ is taken $k-1$ times. For example, $A^{(4)}(0)=\mathsf{E}(A)$ and for $k\geqslant 2$ one has

A finite set $\Lambda\subseteq{\mathbf{G}}$ is called dissociated if any equality of the form

implies $\varepsilon_{\lambda}=0$ for all $\lambda\in\Lambda$. Let ${\rm dim}(A)$ be the size of the largest dissociated subset of $A$ and we call ${\rm dim}(A)$ the additive dimension of $A$. Let us denote all combinations of the form $\{\sum_{\lambda\in\Lambda}\varepsilon_{\lambda}\lambda\}_{\varepsilon_{\lambda}\in\{0,\pm 1\}}$ as ${\rm Span\,}(\Lambda)$.

We need the generalized triangle inequality [25, Theorem 7].

The signs $\ll$ and $\gg$ are the usual Vinogradov symbols. All logarithms are to base $e$.

In this section we follow the well–known logic (see [10]) and first consider the simplest case ${\mathbf{G}}=\mathbb{F}_{2}^{n}$, where the technical difficulties are minimal, and the case of general groups will be considered later.

We need a generalization of a result from [27, Remark 6] (also see the proof of [27, Theorem 4]). For the convenience of the reader, we recall our argument.

Now we are ready to prove our main technical proposition.

We derive some consequences of Proposition 6. Let us start with the case when ${\mathcal{M}}(A)$ is really small, namely, ${\mathcal{M}}^{2}(A)\leqslant(2-\varepsilon)|A|^{2}/K$.

Now we are ready to obtain our structural result for sets with small doubling and small ${\mathcal{M}}(A)$. Given two sets $A,B\subseteq{\mathbf{G}}$ we write $A\dotplus B$ if $|A+B|=|A||B|$.

The following consequence of Proposition 6 allows us to have a “regularization” of any set $A$ in terms of the doubling constant of $A$ and its density, see inequality (40) below. Given a set $A\subseteq{\mathbf{G}}$ put $K[A]:=|A-A|/|A|$.