THE LARGEST (k,ℓ)-SUM-FREE SUBSETS YIFAN JING AND SHUKUN WU Dedicated to the memory of Jean Bourgain Abstract. Let M(2,1)(N) be the infimum of the largest sum-free subset of any set of N positive integers. An old conjecture in additive combinatorics asserts that there is a constant c = c(2, 1) and a function ω(N) as N , such that → ∞ → ∞ cN + ω(N) < M(2,1)(N) < (c + o(1))N. The constant c(2, 1) is determined by Eberhard, Green, and Manners, while the existence of ω(N) is still wide open. In this paper, we study the analogous conjecture on (k,ℓ)-sum-free sets and restricted (k,ℓ)-sum-free sets. We determine the constant c(k,ℓ) for every (k,ℓ)- sum-free sets, and confirm the conjecture for infinitely many (k,ℓ). 1. Introduction In 1965, Erd˝os asked the following question [11]. Given an arbitrary sequence A of N different positive integers, what is the size of the largest sum-free subsequence of A? By sum-free we mean that if x, y, z A, then x + y = z. Let ∈ 6 M(2,1)(N) = inf max S . A⊆N>0 S⊆A | | |A|=N S is sum-free Using a beautiful probabilistic argument, Erd˝os showed that every N-element set >0 A N contains a sum-free subset of size at least N/3, in other words, M(2,1)(N) N/⊆3. It turns out that it is surprisingly hard to improve upon this bound. The≥ result was later improved by Alon and Kleitman [2], who showed that M(2,1)(N) (N +1)/3. Bourgain [7], using an entirely different Fourier analytic argument, showed≥ that M(2,1)(N) (N + 2)/3, which is the best lower bound on M(2,1)(N) to date. arXiv:2001.05632v3 [math.CO] 11 Jan 2021 In particular, the≥ following conjecture has been made in a series of papers. See [11, 7, 10, 28] for example. Conjecture 1. There is a function ω(N) as N , such that →∞ →∞ N M (N) > + ω(N). (2,1) 3 2010 Mathematics Subject Classification. Primary 11B30; Secondary 11K70, 05D10. 1 THE LARGEST (k,ℓ)-SUM-FREE SUBSETS 2 On the other hand, a recent breakthrough by Eberhard, Green, and Manners [10] proved that M(2,1)(N)=(1/3+ o(1))N. More precisely, they showed that for every ε > 0, when N is large enough, there is a set A N>0 of size N, such that every subset of A of size at least (1/3+ ε)N contains x,⊆ y, z with x + y = z. This result is one of the first beautiful applications of the arithmetic regularity lemma. Later, using a completely different argument, the result is generalized by Eberhard [9] to k k-sum-free set. A set A is k-sum-free if for every y, x1,...,xk A, y = i=1 xi. Eberhard proved that for every ε > 0, there is a set A N>0 of∈ size N,6 such that every subset of A of size at least (1/(k +1)+ ε)N contains⊆ a k-sum. ForP more background we refer to the survey [28]. In this paper, we study the analogue of the Erd˝os sum-free set problem for (k,ℓ)- sum-free sets. Given two positive integers k,ℓ with k>ℓ, a set A is (k,ℓ)-sum-free if for every x ,...,x ,y ,...,y A, k x = ℓ y . For example, using the 1 k 1 ℓ ∈ i=1 k 6 j=1 j notation of (k,ℓ)-sum-free, sum-free is (2, 1)-sum-free; k-sum-free is (k, 1)-sum-free. Finding largest (k,ℓ)-sum-free sets inP some givenP structures is well-studied in the past fifty years, for example, the size of the maximum (k,ℓ)-sum-free sets in finite cyclic groups was determined recently by Bajnok and Matzke [4], and the size in compact abelian groups was determined by Kravitz [18]. For every A N>0, let ⊆ M(k,ℓ)(A) = max S , and M(k,ℓ)(N) = inf M(k,ℓ)(A). S⊆A | | A⊆N>0 S is (k,ℓ)-sum-free |A|=N The problem of determining M(k,ℓ)(N) is suggested by Bajnok [3, Problem G.41]. In fact, we can also make the following conjecture for (k,ℓ)-sum-free set, which is an analogue of Conjecture 1. Conjecture 2. Let k>ℓ> 0. There is a constant c = c(k,ℓ) > 0, and a function ω(N) as N , such that →∞ →∞ cN + ω(N) < M(k,ℓ)(N) < (c + ε)N, for every ε> 0. As we mentioned above, the constant c(k,ℓ) in Conjecture 2 for (k,ℓ)=(2, 1) is de- termined by Eberhard, Green, and Manners [10], and for (k,ℓ)=(k, 1) is determined by Eberhard [9]. The conjecture for (k,ℓ)=(3, 1) is confirmed by Bourgain [7]. Our first result determines the constant c(k,ℓ) in Conjecture 2 for every (k,ℓ) (see statements (i) and (iv) of Theorem 1.1), which answers a question asked by Bajnok [3] when the ambient group is Z. The statement (ii) of Theorem 1.1 also confirms Conjecture 2 for infinitely many (k,ℓ). Theorem 1.1. Let k,ℓ be two positive integers and k>ℓ. Then the following hold: THE LARGEST (k,ℓ)-SUM-FREE SUBSETS 3 (i) for every k,ℓ, we have M (N) N . (k,ℓ) ≥ k+ℓ (ii) suppose k =5ℓ. Then N log N (1) M (N) + c , (k,ℓ) ≥ k + ℓ log log N where c> 0 is an absolute constant that only depends on k,ℓ. (iii) for every set A of N positive integers, for every positive even integer u, there is an odd integer v<u such that if k =(u + v)ℓ/(u v), then − N log N (2) M (A) + c , (k,ℓ) ≥ k + ℓ log log N where c> 0 is an absolute constant that only depends on k,ℓ. M 1 (iv) for every k,ℓ, we have (k,ℓ)(N)= k+ℓ + o(1) N. We remark that Theorem 1.1 (iii) also implies estimate (1) when k = 3ℓ, which in particular covers the (3, 1)-sum-free case obtained by Bourgain. This is because when u = 2, the only possible value of v is 1, and this gives us k =3ℓ. It follows that estimate (2) holds for every N-element set A when k =3ℓ. Hence, by the definition of M(k,ℓ)(N), we prove estimate (1) when k =3ℓ. The upper bound construction given by Eberhard, Green, and Manners [10] for (2, 1)-sum-free set actually works in a more general setting: restricted (2, 1)-sum-free set. A set A is restricted (k,ℓ)-sum-free if for every k distinct elements a1,...,ak in A, and ℓ distinct elements b ,...,b in A, we have k a = ℓ b . Let 1 ℓ i=1 i 6 j=1 j M(k,ℓ)(N) = inf maxP SP. A⊆N>0 S⊆A | | |A|=N S is restricted (k,ℓ)−sum free c Clearly, we have that M (N) M (N). Our next theorem gives us an upper (k,ℓ) ≤ (k,ℓ) bound on M(k,ℓ)(N) when k 2ℓ + 1. ≤ c Theorem 1.2. Let k,ℓ be positive integers, and k 2ℓ +1. Then c ≤ 1 M (N)= + o(1) N. (k,ℓ) k + ℓ Overview. The paper is organizedc as follows. In the next section, we provide some basic definitions and properties in additive combinatorics, harmonic analysis, and model theory (or more precisely, nonstandard analysis) used later in the proof. In Section 3, we prove a variant of the weak Littlewood conjecture, based on the ideas introduced by Bourgain [7]. Theorem 1.1 (i) is proved by using the probabilistic argument introduced by Erd˝os, and some structural results for the (k,ℓ)-sum-free open set on the torus. This is included in Section 4. One of the main parts of the THE LARGEST (k,ℓ)-SUM-FREE SUBSETS 4 paper is to prove Theorem 1.1 (ii) and (iii). The special case for (3, 1)-sum-free set is proved by Bourgain [7], but his argument relies heavily on the fact that a certain term of the Fourier coefficient of the characteristic function is multiplicative, which is not true for the other (k,ℓ). Here we introduce a different sieve function, as well as a finer control on the functions we constructed. We will discuss it in detail in Section 5. In Sections 6 and 7, we prove Theorem 1.1 (iv). The proof goes by showing that the constructions given by Eberhard [9] for (k, 1)-sum-free sets, the Følner sequence, is still the correct construction for the other (k,ℓ)-sum-free sets. The new ingredients contain structural results for the large infinite (k,ℓ)-sum-free sets, which can be viewed as a generalization of theLuczak–Schoe n Theorem [21]. We will prove Theorem 1.2 in Section 8. In Section 9, we make some concluding remarks, and pose some open problems. 2. Preliminaries 2.1. Additive combinatorics. Throughout the paper, we use standard definitions and notation in additive combinatorics as given in [27]. Let p be a prime, and let m, n, N ranging over positive integers. Given a, b, N N and a < b, let [a, b] := [a, b] N, and let [N] := [1, N].