Hilbert cubes in progression-free sets and in the set of squares

Abstract

Let S 2 be the set of integer squares. We show that the dimension d of a Hilbert cube a 0 + {0, a 1 } + · · · + {0, a d } ⊂ S 2 is bounded by d = O(log log N ). Letdenote a Hilbert cube of dimension d. Brown, Erdős and Freedman [2] asked whether the maximal dimension of a Hilbert cube in the set of squares is absolutely bounded or not. Experimentally, one finds only very small cubes such asObserve that in this example 29 2 and 37 2 occur as sums in two different ways. Cilleruelo and Granville [3] and Solymosi [15] and Alon, Angel, Benjamini and Lubetzky [1] explain that the Bombieri-Lang conjecture implies that d is absolutely bounded. Hegyvári and Sárközy ([11], Theorem 1) proved that for the set of integer squares S 2 ∩ [1, N ] the maximal dimension is bounded by d = O((log N ) 1/3 ). Dietmann and Elsholtz ([7], Theorem 3) improved this to d = O((log log N ) 2 ). Here we further reduce that bound.Theorem 1 (Main theorem). Let S 2 denote the set of integer squares. Let N be sufficiently large, let a 0 be a non-negative integer and let A = {a 1 , . . . , a d } be a set of distinct positive integers such that H(a 0 ; a 1 , . . . , a d ) ⊆ S 2 ∩ [1, N ]. Then d ≤ 7 log log N.A comparable bound was proved in the author's earlier paper ([7], Theorem 1) for higher powers instead of squares.Remark. The special case of subsetsums, i.e. Hilbert cubes with a 0 = 0, was previously studied by Csikvári ([5], Corollary 2.5), who proved in this case the same bound d = O(log log N ). His method of proof would not extend to the general case of a 0 = 0.Corollary 1. Let f (x) = ax 2 + bx + c be a quadratic polynomial where a, b, c ∈ Z such that a > 0, and let S = {f (x) : x ∈ N}. Let a 0 be a non-negative integer and

Hilbert cubes in arithmetic sets

“…A comparable bound was proved in the authors' earlier paper ( [11], Theorem 1) for k-th powers (k ≥ 3) instead of squares. The treatment of higher powers was easier for the following reason: By a deep theorem of Darmon and Merel [9], following the proof of Fermat's last theorem, for k ≥ 3 there are no 3-progressions in the set of k-th powers.…”

“…Remark. With regard to our earlier result, Noga Alon kindly pointed out to us that Lemma 5 of [11], which corresponds to Lemma 9 in this paper, is actually a version of a result of Erdős and Rado [14] on ∆-systems (or sunflowers). On this subject small quantitative improvements are due to Kostochka [35], even though the explicit dependence on the parameters h and v is not well understood, and apparently at least one of the parameters h and v is fixed.…”

“…In this paper we introduce a tool, Lemma 1 below, to the study of Hilbert cubes, which extends a recent method of Schoen [43], and which eventually achieves exponential sumset growth for the sets under consideration. For the set of squares and progression-free sets (Theorems 1 and 2) this improves, and simplifies, our work in [11]. The two methods above make use of global properties of the set S, such as avoiding arithmetic progressions.…”

“…Note that the assumption 4a + b 2 − 4ac ≥ 1 is not really necessary, but simplifies the proof as it avoids possibly hitting the number zero which is excluded in our definition of the set S 2 of squares. The method of proof allows us also to prove the following theorem, which improves [11,Theorem 2]. The special case of subset sums has been studied by Schoen [43,…”

“…Hegyvári and Sárközy ([30], Theorem 1) proved that for the set of integer squares S 2 ∩ [1, N ] the maximal dimension of a Hilbert cube is bounded by d = O((log N ) 1/3 ). In a previous paper ( [11], Theorem 3) we improved this to d = O((log log N ) 2 ). Here we further reduce that bound.…”