The growth rate of tri-colored sum-free sets

Abstract

Let G be an abelian group. A tri-colored sum-free set in G is a collection of triples (a a a i , b b b i , c c c i ) in G such that a a a i + b b b j + c c c k = 0 if and only if i = j = k. Fix a prime q and let C q be the cyclic group of order q. Let θ = min ρ>0 (1 + ρ + · · · + ρ q−1 )ρ −(q−1)/3 . Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (building on previous work of Croot, Lev and Pach, and of Ellenberg and Gijswijt) showed that a tri-colored sum-free set in C n q has size at most 3θ n . Between this paper and a paper of Pebody, we will show that, for any δ > 0, and n sufficiently large, there are tri-colored sum-free sets in C n q of size (θ − δ ) n . Our construction also works when q is not prime. IntroductionLet G be an abelian group. Let t t t ∈ G n . We make the following slightly nonstandard definition: a sum-free set in G n with target t t t is a collection of triples (a a ato make the target 0 0 0 (as we did in the abstract, and as is more standard), but allowing an arbitrary target will simplify our notation. The usual terminology is "tri-colored sum-free set", but we omit the reference to the coloring as we never consider any other kind.If X ⊂ G n is a set with no three-term arithmetic progressions, then {(x x x, x x x, −2x x x) : x x x ∈ X} is sum-free with target 0 0 0, so lower bounds on sets without three-term arithmetic progressions are also bounds on sum-free sets. The reverse does not hold: the largest known three-term arithmetic progression free subsets of C n 3 (where C q is the cyclic group of order q) are of size 2.217 n [10]. Before this paper, the largest known sum-free sets in C n 3 were of size 2.519 n [1]; this paper will raise the bound to 2.755 n and show that this bound is tight. Letting r 3 (G n ) denote the largest subset of G n with no three-term arithmetic progressions, the question of whether lim sup n→∞ r 3 (G n ) 1/n < |G| was open, until recently, for every abelian G containing elements of order greater than two. The breakthrough work of Croot, Lev, and Pach [9] introduced a polynomial method to prove that strict inequality holds when G is cyclic of order 4, and Ellenberg and Gijswijt [12] built upon their ideas to prove it for cyclic groups of odd prime order. Blasiak et al. [5] applied the same method to prove upper bounds for sum-free sets in G n for any fixed finite abelian group G.We recall here one case of their bound. Let C q be the cyclic group of order q. Let θ = min β >0 (1 + β + · · · + β q−1 )β −(q−1)/3 and let ρ be the value of β at which the minimum is attained. We note that the minimum is attained at a unique point which belongs to (0, 1) because (1 + β + · · · + β q−1 )β −(q−1)/3 approaches ∞ as

A lower bound for the k‐multicolored sum‐free problem in Zmn

“…The conjecture in the first version of the paper of Kleinberg, Sawin, and Speyer stated that for every , the probability distribution occurs as the marginal of an ‐symmetric probability distribution on . As mentioned above, this was proved by Pebody .…”

“…After Blasiak et al . established the upper bound, Kleinberg, Sawin, and Speyer proved Theorem for and any . Their proof uses a statement that had been formulated as a conjecture in an earlier version of their paper and was then proved by Pebody .…”

“…It is worth noting that the proof of Theorem is not a straightforward generalization of the work of Kleinberg, Sawin, and Speyer for the case . Their argument uses a random sampling process to find a large 3‐colored sum‐free set within a certain collection of 3‐tuples.…”

“…Their argument uses a random sampling process to find a large 3‐colored sum‐free set within a certain collection of 3‐tuples. Although our approach for proving Theorem is the same as in , serious challenges arise with the probabilistic sampling argument. The main difficulty in generalizing the work in arises in proving Proposition , which states that there cannot be too many special pairs of ‐tuples with increased conditional probabilities in the sampling process.…”

“…The most crucial tool for the proof of the proposition for is an entropy inequality that we will introduce in Section 4. Furthermore, in the case of , one also only needs a much weaker version of Proposition , and the corresponding argument is just a single paragraph in .…”