The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant $c > 0$. We explore what structure $A$ must have if $\| \widehat{1_A}\|_1\leq K\log N$ for some constant $K$. Under such an assumption we prove, for instance, that $A$ contains a subset $A'\subseteq A$ with $\lvert A\rvert \geq N^{0.99}$ such that $\lvert A'+A'\rvert \ll K^{O(1)}\lvert A'\rvert$. As a consequence, for any $k\geq 3$, if $N$ is sufficiently large depending on $k$ and $K$, then $A$ must contain an arithmetic progression of length $k$. A byproduct of our analysis is a (slightly) improved bound for the constant $c$.
For a finite set A ⊂ Z the Fourier transform 1 A : R/Z → C is defined by 1 A (θ) = n∈A e(-nθ), where e(x) := e 2πix . Define
A famous conjecture of Littlewood stated that1 ∥ 1 A ∥ 1 ≫ log N for any set A of N integers, which is achieved for example by an arithmetic progression of length N . This was proved in 1981 independently by Konyagin [11] and McGehee, Pigno, and Smith [12]. A discussion of the problem and its history may be found in [3,Chapter 10].
This lower bound is universal, yet rarely achieved -in fact generically we expect ∥ 1 A ∥ 1 to be closer to the trivial upper bound N 1/2 , which is an immediate consequence of the Cauchy-Schwarz inequality and Parseval’s identity. This is achieved when A is lacunary, for example, or a random positive density subset of an interval (with high probability). In fact, the only examples we know that achieve O(log N ) are arithmetic progressions and simple variations, for example the union of O(1) arithmetic progressions and some small unstructured set.
It is therefore natural, especially from the viewpoint of modern additive combinatorics, to ask the following inverse question. Question 1.1. Let K > 0 be some large constant. What can we say about the structure of sets A ⊂ Z with |A| = N and ∥ 1 A ∥ 1 ⩽ K log N ? This question was asked by the second author [6] and one possible precise conjecture in this direction was formulated by Petridis [13,Question 5.1] (see Section 6 for further details). While we are some way from a full resolution of this question, in this paper we offer the following weak partial progress.
Recall that the additive energy E(B) of a finite set B ⊂ Z is defined by
(1.1)
We have
as can be checked using the orthogonality relation for characters. We also write ω[B] := E(B)/|B| 3 for the normalised additive energy of B, which satisfies 0 < ω[B] ⩽ 1.
Theorem 1.2. Let N be a sufficiently large positive integer. Let δ ∈ (0, 1 2 ] and
The set A ′ produced in Theorem 1.2 is in fact an initial segment of A . The following two corollaries are almost immediate consequences of this and standard results in additive combinatorics.
Corollary 1.3. Let N be a sufficiently large positive integer. Let K > 0 and suppose that A ⊂ Z is a finite set of size N such that ∥ 1 A ∥ 1 ⩽ K log N . Then there exists an arithmetic progression P of length
where c K > 0 depends only on K.
Proof. Apply Theorem 1.2 with δ = 1 2 . Applying the Balog-Szemerédi-Gowers theorem (see, for example, [16,Theorem 2.31]) to the resulting set A ′ , we obtain a set
By the Freiman-Ruzsa theorem (see, for example [16,Theorem 5.33]), there are arithmetic progressions P 1 , . . . , P r with r = O K (1) such that A ′′ ⊆ P 1 + • • • + P r and
By averaging there exists some i, 1 ⩽ i ⩽ r, and x such that
Since N is large, the length of P i is bounded below by N c K for some c K > 0. □ Corollary 1.4. Let K > 0 and let k ⩾ 3 be an integer. Let N be sufficiently large in terms of k, K and let A ⊂ Z be a finite set of size
Then A contains an arithmetic progression of length k.
Proof. This follows immediately from Corollary 1.3 and Szemerédi’s theorem. □
The key feature of Theorem 1.2 is that the lower bound on E(A ′ ) is a constant times |A ′ | 3 . If one is prepared to tolerate logarithmic losses then a one line application of Hölder’s inequality on the Fourier side shows that we in fact have E(A) ⩾ N 3 /(K log N ) 2 . However, this would certainly be too weak to deduce Corollary 1.4 given our current knowledge of bounds in Szemerédi’s theorem when k ⩾ 4.
1.1. Previous structural results. We now summarise what was previously known concerning this inverse question for sets with small ∥ 1 A ∥ 1 .
• As noted above, it is a trivial consequence of Hölder’s inequality that A itself must have reasonably large additive energy: Lemma 3] noted that an argument of Zygmund [18,ter XII, 7.6] implies that the largest dissociated 2 subset of A has size O K ((log N ) 3 ). This was independently rediscovered by Bedert [1]. • Petridis [13] showed that A cannot have genuine 3-dimensional structure, in a certain precise sense. • Picking up on the theme of Petridis, Hanson [8] showed that A cannot have genuine 2-dimensional structure, again in a sense he was able to make precise. • If we replace the upper bound ⩽ K log N with ⩽ K, and replace Z with a finite abelian group G, then a much more satisfactory answer is known: the second author and Sanders [7] have shown that if A ⊆ G has ∥ 1 A ∥ 1 ⩽ K then 1 A can be written as a simple linear combination of O K (1) indicator functions of cosets of subgroups of G. This condition is (qualitatively) necessary and sufficient.
Part of the motivation for inverse results of this nature originated in work of Bourgain [2], which highlighted a connection between Littlewood’s conjecture and the task of improving the lower bounds for the sum-free subset problem in additive combinatorics. Bedert [1] has made significant progress on this problem recently, making use
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