Generalizations of Product-Free Subsets

In this paper, we present some generalizations of Gowers’s result about product-free subsets of groups. For any group G of order n, a subset A of G is said to be product-free if there is no solution of the equation ab = c with a, b, c ∈ A. Previous results showed that the size of any product-free subset of G is at most n/δ1/3, where δ is the smallest dimension of a nontrivial representation of G. However, this upper bound does not match the best lower bound. We will generalize the upper bound to the case of product-poor subsets A, in which the equation ab = c is allowed to have a few solutions with a, b, c ∈ A. We prove that the upper bound for the size of product-poor subsets is asymptotically the same as the size of product-free subsets. We will also generalize the concept of product-free to the case in which we have many subsets of a group, and different constraints about products of the elements in the subsets. 1. Background Let G be a group. A subset S of G is product-free if there do not exist a, b, c ∈ S (not necessarily distinct) such that ab = c. One can ask about the existence of large product-free subsets for various groups, such as the groups of integers or compact topological groups. Assume that G is a finite group of order n > 1. Let α(G) denote the size of the largest product-free subset of G, and β(G) = α(G)/n. The purpose is to find good bounds for α(G) and β(G) as a function of G, or as a function of n. If G is abelian, this problem was solved by Green and Ruzsa in 2005 [3]. They gave an exact value of α(G) as a function of some characteristics of the abelian group G. However, the problem is much harder for nonabelian G. The first appearance of the problem of computing α(G) for nonabelian G seems to have been in a 1985 paper [2]. The construction of product-free subsets given in [2] is quite simple: if H is a proper subgroup of G, then any nontrivial coset ofH is product-free. Therefore, a lower bound for β(G) can be derived: β(G) ≥ m, where m is the index of the largest proper subgroup of G. In 1997, Kedlaya [5] improved this bound to cm for some constant c by showing that if H has index k then one can in fact find a union of ck cosets of H that is product-free. This gives the best known lower bound on α(G) for general G. 2000 Mathematics Subject Classification. Primary 20D60; secondary 20P05. The first author was supported by NSF CAREER grant DMS-0545904 and a Sloan Research Fellowship. The second author was supported by the Paul E. Gray (1954) Endowed Fund for the MIT Undergraduate Research Opportunities Program. 1 2 KIRAN S. KEDLAYA AND XUANCHENG SHAO Theorem 1. β(G) ≥ cm for some constant c. On the other hand, Gowers recently established a nontrivial upper bound for α(G) using a remarkably simple argument [4] (see also [6] in this volume). The strategy of Gowers is to consider three sets A,B,C for which there is no solution of the equation ab = c with a ∈ A, b ∈ B, c ∈ C, and give an upper bound on |A| · |B| · |C|, where |A| denote the order of the set A. Theorem 2. If A, B, C are subsets of G such that there is no solution of the equation ab = c with a ∈ A, b ∈ B, c ∈ C, then |A| · |B| · |C| ≤ n/δ, where δ is defined as the smallest dimension of a nontrivial representation of G. Consequently, β(G) ≤ δ. To prove this, a certain bipartite Cayley graph Γ associated to G is constructed. The vertex set of Γ is V1 ∪ V2, where each Vi is a copy of G, with an edge from x ∈ V1 to y ∈ V2 if and only if yx ∈ A. Therefore, there are no edges between B ⊆ V1 and C ⊆ V2. Let N be the incidence matrix of Γ, with columns indexed by V1 and rows by V2, with an entry in row x and column y if xy is an edge of Γ. Define M = NN; then the largest eigenvalue of M is |A|. Let λ denote the second largest eigenvalue of the matrix M . Gowers’s theorem relies on the following two lemmas. Lemma 3. The second largest eigenvalue λ of the matrix M is at most n · |A|/δ. Lemma 4. |A| · |B| · |C| ≤ n · λ |A| . It can be proved that the group G has a proper subgroup of index at most cδ for some constant c [7]. Therefore, we have the following bounds for α(G). Theorem 5. cn/δ ≤ α(G) ≤ cn/δ for some constant c. Since the gap between the lower bound and the upper bound for α(G) appears quite small, one might ask about closing it. However, it has been proved in [6] that Gowers’s argument alone is not sufficient since the upper bound in Theorem 2 cannot be improved if the three sets A, B, and C are allowed to be different. In addition, Gowers also made some generalizations to the case of many subsets. Instead of finding two elements a and b in two subsets A and B such that their product is in a third subset C, he proposed to find x1, . . . , xm in m subsets such that for every nonempty subset F ⊂ {1, 2, . . . ,m}, the product of those xi with i ∈ F lies in a specified subset. In this paper, instead of trying to close the gap between the lower bound and the upper bound for α(G), we will show that Gowers’s Theorem 2 can actually be generalized to product-poor subsets of a group. We will give the precise definition of product-poor subsets as well as the upper bound for the size of product-poor subsets in Section 2. In Section 3, we will examine Gowers’s generalization to the case of many subsets, and we will further generalize it to the problem of finding x1, . . . , xm in m subsets such that for certain (not all) subsets F ⊂ {1, 2, . . . ,m} the product of those xi with i ∈ F lies in a specified subset. 2. Product-poor subsets of a group In this section, we will state and prove a generalization of Theorem 2. We will consider the size of the largest product-poor subset instead of product-free subset. GENERALIZATIONS OF PRODUCT-FREE SUBSETS 3 In product-poor subsets, there are a few pairs of elements whose product is also in this set. It turns out that we can derive the same asymptotic upper bound for the size of a product-poor subset. Despite the fact that the best known lower bound and the upper bound for the size of the largest product-free subset do not coincide, these two bounds do coincide asymptotically for the largest product-poor subset. First of all, we give the precise definition of a product-poor subset. Definition 6. A subset A of group G is p-product-poor iff the number of pairs (a, b) ∈ A×A such that ab ∈ A is at most p|A|2. We now give a generalization of Gowers’s argument. We change the condition that there are no solutions of the equation ab = c with a ∈ A, b ∈ B, c ∈ C, to the weaker condition that there are only a few solutions of that equation. In fact, Babai, Nikolov, and Pyber [1] recently discovered a more general result depending on probability distributions in G. However, here we want to emphasize the result that the upper bound of |A| · |B| · |C| is asymptotically the same for both productfree subsets and product-poor subsets. Note that this theorem is similar to a lemma in [4], which is stated in Lemma 10 below. Theorem 7. Let G be a group of order n. Let A, B, and C be subsets of G with orders rn, sn, and tn, respectively. If there are exactly prstn solutions of the equation ab = c with a ∈ A, b ∈ B, c ∈ C, then rst(1 − p)δ ≤ 1. Proof. Let v be the characteristic function of B, and put w = v − s1. Then w · 1 = 0, w ·w = (1− s) · sn+ s(n− sn) = s(1 − s)n ≤ sn, so by Lemma 3, ‖Nw‖2 ≤ rnsn/δ. Let N be the incidence matrix of G of size n × n, in which there is an entry in row x and column y iff xy ∈ B. Consider the submatrix N1 of N containing those rows corresponding to the elements in C, and those columns corresponding to the elements in A. The matrix N1 has size tn× rn. Suppose that there are ki ones in row i of N1 (1 ≤ i ≤ tn). Note that there exists a one-to-one correspondence between the solutions of the equation ab = c with a ∈ A, b ∈ B, c ∈ C, and the nonzero entries in N1. As a result, we have k1 + k2 + . . .+ ktn = prstn . Using the above equality, we have ‖Nw‖ ≥ (k1 − rsn) + (k2 − rsn) + . . .+ (ktn − rsn)