Abelian sections of the symmetric groups with respect to their index

We show the existence of an absolute constant $$\alpha >0$$ α > 0 such that, for every $$k \ge 3$$ k ≥ 3 , $$G:= \mathop {\mathrm {Sym}}(k)$$ G : = Sym ( k ) , and for every $$H \leqslant G$$ H ⩽ G of index at least 3, one has $$|H/H'| \le |G:H|^{\alpha / \log \log |G:H|}$$ | H / H ′ | ≤ | G : H | α / log log | G : H | . This inequality is the best possible for the symmetric groups, and we conjecture that it is the best possible for every family of arbitrarily large finite groups.

The Quest for Strong Inapproximability Results with Perfect Completeness

The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

A Subexponential Upper Bound for van der Waerden Numbers $W(3,k)$

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then $$N\le \exp(O(k^{1-c}))\,.$$

A UNIFORM BOUND ON THE OPERATOR NORM OF SUB-GAUSSIAN RANDOM MATRICES AND ITS APPLICATIONS

For an $N \times T$ random matrix $X(\beta )$ with weakly dependent uniformly sub-Gaussian entries $x_{it}(\beta )$ that may depend on a possibly infinite-dimensional parameter $\beta \in \mathbf {B}$ , we obtain a uniform bound on its operator norm of the form $\mathbb {E} \sup _{\beta \in \mathbf {B}} ||X(\beta )|| \leq CK \left (\sqrt {\max (N,T)} + \gamma _2(\mathbf {B},d_{\mathbf {B}})\right )$ , where C is an absolute constant, K controls the tail behavior of (the increments of) $x_{it}(\cdot )$ , and $\gamma _2(\mathbf {B},d_{\mathbf {B}})$ is Talagrand’s functional, a measure of multiscale complexity of the metric space $(\mathbf {B},d_{\mathbf {B}})$ . We illustrate how this result may be used for estimation that seeks to minimize the operator norm of moment conditions as well as for estimation of the maximal number of factors with functional data.

A variant of the Corners theorem

The Corners theorem states that for any α > 0 there exists an N0 such that for any abelian group G with |G| = N ≥ N0 and any subset A ⊂ G×G with |A| ≥ αN 2 we can find a corner in A, i.e. there exist x, y, d ∈ G with d ≠ 0 such that (x,y),(x+d,y),(x,y+d) ∈ A. Here, we consider a stronger version, in which we try to find many corners of the same size. Given such a group G and subset A, for each d ∈ G we define S d ={(x,y) ∈ G × G: (x,y),(x+d,y),(x,y+d) ∈ A}. So |S d | is the number of corners of size d. Is it true that, provided N is sufficiently large, there must exist some d ∈G\{0} such that |S d |>(α3-ϵ)N2? We answer this question in the negative. We do this by relating the problem to a much simpler-looking problem about random variables. Then, using this link, we show that there are sets A with |S d |>Cα3.13N2 for all d ≠ 0, where C is an absolute constant. We also show that in the special case where $G = {\mathbb{F}}_2^n$ , one can always find a d with |S d |>(α4-ϵ)N2.

Difference sets in higher dimensions

Let d ≥ 3 be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have $$\begin{equation*} |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\end{equation*}$$ where δ > 0 is an absolute constant only depending on d. This improves upon an earlier result of Freiman, Heppes and Uhrin, and makes progress towards a conjecture of Stanchescu.

Near-perfect clique-factors in sparse pseudorandom graphs

We prove that, for any $t \ge 3$ , there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of k t covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.

On the dimensional weak-type (1,1) bound for Riesz transforms

Let [Formula: see text] denote the [Formula: see text] Riesz transform on [Formula: see text]. We prove that there exists an absolute constant [Formula: see text] such that [Formula: see text] for any [Formula: see text] and [Formula: see text], where the above supremum is taken over measures of the form [Formula: see text] for [Formula: see text], [Formula: see text], and [Formula: see text] with [Formula: see text]. This shows that to establish dimensional estimates for the weak-type [Formula: see text] inequality for the Riesz transforms it suffices to study the corresponding weak-type inequality for Riesz transforms applied to a finite linear combination of Dirac masses. We use this fact to give a new proof of the best known dimensional upper bound, while our reduction result also applies to a more general class of Calderón–Zygmund operators.

On computing the Lyapunov exponents of reversible cellular automata

We consider the problem of computing the Lyapunov exponents of reversible cellular automata (CA). We show that the class of reversible CA with right Lyapunov exponent 2 cannot be separated algorithmically from the class of reversible CA whose right Lyapunov exponents are at most $$2-\delta$$ 2 - δ for some absolute constant $$\delta >0$$ δ > 0 . Therefore there is no algorithm that, given as an input a description of an arbitrary reversible CA F and a positive rational number $$\epsilon >0$$ ϵ > 0 , outputs the Lyapunov exponents of F with accuracy $$\epsilon$$ ϵ . We also compute the average Lyapunov exponents (with respect to the uniform measure) of the reversible CA that perform multiplication by p in base pq for coprime $$p,q>1$$ p , q > 1 .

Character Ratios for Exceptional Groups of Lie Type

We prove character ratio bounds for finite exceptional groups $G(q)$ of Lie type. These take the form $\dfrac{|\chi (g)|}{\chi (1)} \le \dfrac{c}{q^k}$ for all nontrivial irreducible characters $\chi$ and nonidentity elements $g$, where $c$ is an absolute constant, and $k$ is a positive integer. Applications are given to bounding mixing times for random walks on these groups and also diameters of their McKay graphs.