We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]: suppose we are given a finite set of places S over Q containing the archimedean place and excluding the prime 2, an irrational isotropic form q of rank n ≥ 4 on QS, a product of p-adic intervals Ip, and a product Ω of star-shaped sets.We show that unless n = 4 and q is split in at least one place, the number of S-integral vectors v ∈ TΩ satisfying simultaneously q(v) ∈ Ip for p ∈ S is asymptotically given by λ(q, Ω)|I| · T n−2 ,as T goes to infinity, where |I| is the product of Haar measures of the p-adic intervals Ip.The proof uses dynamics of unipotent flows on S-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an S-arithmetic variant of the α-function introduced in [10], and an S-arithemtic version of a theorem of Dani-Margulis [7].The proof of Theorem 1.5 rests upon a number of ingredients. We will use the dynamics on G/Γ, where G = SL n (Q S ) and Γ = SL n (Z S ), which can be identified with a set L S of unimodular S-lattices. First, we will relate the counting problem to a question about the asymptotic behavior of integrals of the form K f (a t k∆)dm(k), where K is a maximal compact subgroup of SO(q), x is an element in G/Γ related to q, m is the normalized Haar measure of K, and a t is a 1-parameter diagonal subgroup of SO(q) (see Equation (3) in Section 2.3). Here, f is the Siegel transform of a compactly supported function f defined on Q n S (see Definition 3.7). Such integrals, when f is replaced by a bounded continuous function can be dealt with using an S-arithmetic version of the results in [7], which we will state and prove in Section 6 of this paper.Since f is unbounded, in order to use the equidistribution result just described, one needs to also control the integral of f (a t k∆) when a t k∆ is far into the cusp. As in [10], this is dealt with using the function α S (see section 3.6), which is an analog of α introduced in the previous subsection. We prove Theorem 1.7. With the notation as above, assume that q is not exceptional. For 0 < s < 2 and for any S-lattice ∆ ∈ L S , sup t≻1 K α S (a t k∆) s dm(k) < ∞.Moreover, the bound is uniform as ∆ varies over a compact subset C of L S .

For a group G, we denote by m faithful (G), the smallest dimension of a faithful complex representation of G. Let F be a non-Archimedean local field with the ring of integers O and the maximal ideal p. In this paper, we compute the precise value of m faithful (G) when G is the Heisenberg group over O/p n . We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over O/p n and many of its subgroups. By a theorem of Karpenko and Merkurjev [7, Theorem 4.1], our result yields the precise value of the essential dimension of the latter finite groups.

Abstract. Inspired by Gowers' seminal paper [6], we will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of SL k (Z/(p n Z)) and Sp 2k (Z/(p n Z)). Our method also delivers a lower bound for the minimal degree of a faithful representation of these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups SL k (Zp) and Sp 2k (Zp). We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F , let QG Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u, w ∈ V form an edge if and only if Q(v − w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present paper, we will prove that for a local field F of characteristic zero, the Borel chromatic number of QG Q is infinite if and only if Q represents zero non-trivially over F . The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009 [6].

Let a finite group G act transitively on a finite set X . A subset S ⊆ G is said to be intersecting if for any s 1 , s 2 ∈ S, the element s −1 1 s 2 has a fixed point. The action is said to have the weak Erdős-Ko-Rado (EKR) property, if the cardinality of any intersecting set is at most |G|/|X |. If, moreover, any maximum intersecting set is a coset of a point stabilizer, the action is said to have the strong EKR property. In this paper, we will investigate the weak and strong EKR property and attempt to classify groups in which all transitive actions have these properties. In particular, we show that a group with the weak EKR property is solvable and that a nilpotent group with the strong EKR property is the direct product of a 2-group and an abelian group of odd order.