Littlewood–Offord 问题关注一个向量集合的子和落入给定凸集的数量。我们提出该问题的一个离散变体，用以估计属于 (0,1)-向量的子和数量。随后，我们利用这一结果确定具有给定秩或余秩的图的最大阶。图 G 的秩是其邻接矩阵 A(G) 的秩，而 G 的余秩是 A(G)+I 的秩。
英文摘要：
Littlewood--Offord Problem concerns the number of subsums of a set of vectors that fall in a given convex set. We present a discrete variation of this problem where we estimate the number of subsums that are $(0,1)$-vectors.
We then utilize this to find the maximum order of graphs with given rank or corank. The rank of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the corank of $G$ is the rank of $A(G)+I$.
Motivated by the inverse Littlewood-Offord problem for linear forms, we study the concentration of quadratic forms. We show that if this form concentrates on a small ball with high probability, then the coefficients can be approximated by a sum of additive and algebraic structures.
Let $\mathbf{v}_i$ be vectors in $\mathbb{R}^d$ and $\{\varepsilon_i\}$ be independent Rademacher random variables. Then the Littlewood-Offord problem entails finding the best upper bound for $\sup_{\mathbf{x} \in \mathbb{R}^d} \mathbb{P}(\sum \varepsilon_i \mathbf{v}_i = \mathbf{x})$. Generalizing the uniform bounds of Littlewood-Offord, Erdős and Kleitman, a recent result of Dzindzalieta and Juškevičius provides a non-uniform bound that is optimal in its dependence on $\|\mathbf{x}\|_2$. In this short note, we provide a simple alternative proof of their result. Furthermore, our proof demonst…
We give a new bound on the probability that the random sum $ξ_1 v_1 +...+ ξ_n v_n$ belongs to a ball of fixed radius, where the $ξ_i$ are iid Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an application, we prove a conjecture of Frankl and Füredi (raised in 1988), which can be seen as the high dimensional version of the classical Littlewood-Offord-Erd\H os theorem.
Let $η_i, i=1,..., n$ be iid Bernoulli random variables. Given a multiset $\bv$ of $n$ numbers $v_1, ..., v_n$, the \emph{concentration probability} $¶_1(\bv)$ of $\bv$ is defined as $¶_1(\bv) := \sup_{x} ¶(v_1 η_1+ ... v_n η_n=x)$. A classical result of Littlewood-Offord and Erd\H os from the 1940s asserts that if the $v_i $ are non-zero, then this probability is at most $O(n^{-1/2})$. Since then, many researchers obtained better bounds by assuming various restrictions on $\bv$. In this paper, we give an asymptotically optimal characterization for all multisets $\bv$ having large concentratio…
We prove several Littlewood-Offord type inequalities for arbitrary groups. In groups having elements of finite order the worst case scenario is provided by the simple random walk on a certain cyclic subgroup. The inequalities we obtain are optimal if the underlying group contains an element of certain order. It turns out that for torsion-free groups Erdős's bound still holds. Our results strengthen and generalize some very recent results by Tiep and Vu.
Let eta_i be iid Bernoulli random variables, taking values -1,1 with probability 1/2. Given a multiset V of n integers v_1,..., v_n, we define the concentration probability as rho(V) := sup_{x} Pr(v_1 eta_1+...+ v_n eta_n=x). A classical result of Littlewood-Offord and Erdos from the 1940s asserts that if the v_i are non-zero, then rho(V) is O(n^{-1/2}). Since then, many researchers obtained improved bounds by assuming various extra restrictions on V. About 5 years ago, motivated by problems concerning random matrices, Tao and Vu introduced the Inverse Littlewood-Offord problem. In the inverse…
In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.
Consider the sum $X(ξ)=\sum_{i=1}^n a_iξ_i$, where $a=(a_i)_{i=1}^n$ is a sequence of non-zero reals and $ξ=(ξ_i)_{i=1}^n$ is a sequence of i.i.d. Rademacher random variables (that is, $\Pr[ξ_i=1]=\Pr[ξ_i=-1]=1/2$). The classical Littlewood-Offord problem asks for the best possible upper bound on the concentration probabilities $\Pr[X=x]$. In this paper we study a resilience version of the Littlewood-Offord problem: how many of the $ξ_i$ is an adversary typically allowed to change without being able to force concentration on a particular value? We solve this problem asymptotically, and present…
Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positive constant $C$. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.
Consider a sum $S_n=v_i\varepsilon_1+\cdots+v_n\varepsilon_{n}$, where $(v_i)^{n}_{i=1}$ are non-zero vectors in $\mathbb{R}^{d}$ and $(\varepsilon_i)^{n}_{i=1}$ are independent Rademacher random variables (i.e., $~{\mathbb{P}(\varepsilon_{i}=\pm 1)=1/2}$). The classical Littlewood-Offord problem asks for the best possible upper bound for $~{\sup_{x}\mathbb{P}(S_n = x)}$. In this paper we consider a non-uniform version of this problem. Namely, we obtain the optimal bound for $\mathbb{P}(S_n = x)$ in terms of the length of the vector $x\in \mathbb{R}^d$.
The classical Erdős-Littlewood-Offord problem concerns the random variable $X = a_1 ξ_1 + \dots + a_n ξ_n$, where $a_i \in \mathbb{R} \setminus \{0\}$ are fixed and $ξ_i \sim \text{Ber}(1/2)$ are independent. The Erdős-Littlewood-Offord theorem states that the maximum possible concentration probability $\max_{x \in \mathbb{R}} \Pr(X = x)$ is $\binom{n}{\lfloor n/2\rfloor} / 2^n$, achieved when the $a_i$ are all $1$. As proposed by Fox, Kwan, and Sauermann, we investigate the general case where $ξ_i \sim \text{Ber}(p)$ instead. Using purely combinatorial techniques, we show that the exact maxim…
Consider a random sum $η_1 v_1 + ... + η_n v_n$, where $η_1,...,η_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $¶(η_1 v_1 + ... + η_n v_n = 0)$ subject to various hypotheses on the $v_1,...,v_n$. In this paper we develop an \emph{inverse} Littlewood-Offord theorem (somewhat in the spirit of Freiman's inverse sumset theorem), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the $v_1,...,v_n$ are efficiently contained in an arithmetic progre…
The classical Erdős-Littlewood-Offord theorem says that for nonzero vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly random $(ξ_1,\dots,ξ_n)\in\{-1,1\}^n$, we have $\Pr(a_1ξ_1+\dots+a_nξ_n=x)=O(n^{-1/2})$. In this paper we show that $\Pr(a_1ξ_1+\dots+a_nξ_n\in S)\le n^{-1/2+o(1)}$ whenever $S$ is definable with respect to an o-minimal structure (for example, this holds when $S$ is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting.
The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable $ε_1 v_1 + \cdots + ε_n v_n$ lies in the Euclidean unit ball, where $ε_1, \ldots, ε_n \in \{-1, 1\}$ are independent Rademacher random variables and $v_1, \ldots, v_n \in \mathbb{R}^d$ are fixed vectors of at least unit length.We extend many known results to the case that the $ε_i$ are obtained from a Markov chain, including the general bounds first shown by Erdős in the scalar case and Kleitman in the vector case, and also under the restriction that the $v_i$ are distinct integers due …
Let $ε_1, \dotsc, ε_n$ be i.i.d. Rademacher random variables taking values $\pm 1$ with probability $1/2$ each. Given an integer vector $\boldsymbol{a} = (a_1, \dotsc, a_n)$, its concentration probability is the quantity $ρ(\boldsymbol{a}):=\sup_{x\in \mathbb{Z}}\Pr(ε_1 a_1+\dots+ε_n a_n = x)$. The Littlewood-Offord problem asks for bounds on $ρ(\boldsymbol{a})$ under various hypotheses on $\boldsymbol{a}$, whereas the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a characterization of all vectors $\boldsymbol{a}$ for which $ρ(\boldsymbol{a})$ is large. In this paper, we stu…
Consider a quadratic polynomial $f\left(ξ_{1},\dots,ξ_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we pro…
If f(x_1, x_2, ..., x_n) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a_1 through a_n, what is the maximum number of sums of the form +/- a_1 +/- a_2 +/-... +/- a_n which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly…
Let $X_1,\ldots,X_n$ be independent identically distributed random vectors in $\mathbb{R}^d$. We consider upper bounds on $\max_x \mathbb{P}(a_1X_1+\cdots+a_nX_n=x)$ under various restrictions on $X_i$ and the weights $a_i$. When $\mathbb{P}(X_i=\pm 1) = \frac {1} {2}$, this corresponds to the classical Littlewood-Offord problem. We prove that in general for identically distributed random vectors and even values of $n$ the optimal choice for $(a_i)$ is $a_i=1$ for $i\leq \frac{n}{2}$ and $a_i=-1$ for $i > \frac {n} 2$, regardless of the distribution of $X_1$. Applying these results for Bernoul…
Let M and N be two r x r matrices over a discrete valuation ring of characteristic zero. The orders (with respect to a uniformizing parameter) of the invariant factors of M form a partition of non-negative integers, called the invariant partition of M. Let the invariant partition of M be mu, of N be nu, and of the product MN be lambda. In this paper we construct a Littlewood-Richardson filling of the skew shape lambda/mu with content nu, and show that this filling is an invariant of the orbit of the pair (M,N) with respect to a natural group action on the pair. We relate the algebraic combinat…
Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=ε_{1}+...+ε_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that \mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}, where A is either an interval of the form [x, \infty) or just a single point. The inequality is exact and the optimal values of c and k are given explicitly. It improves Kwapień's inequality in the case of the Rademacher series. We also provide a new and very short proof of the Littlewood-Offord problem without using Sperner's T…
It is generally hard to count, or even estimate, how many integer points lie in a polytope P. Barvinok and Hartigan have approached the problem by way of information theory, showing how to efficiently compute a random vector which samples the integer points of P with (computable) constant mass, but which may also land outside P. Thus, to count the integer points of P, it suffices to determine the frequency with which the random vector falls in P. We prove a collection of efficiently computable upper bounds on this frequency. We also show that if P is suitably presented by n linear inequalities…
We show that singular numbers (also known as invariant factors or Smith normal forms) of products and corners of random matrices over $\mathbb{Q}_p$ are governed by the Hall-Littlewood polynomials, in a structurally identical manner to the classical relations between singular values of complex random matrices and Heckman-Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of $\text{GL}_N(\mathbb{Z}_p)$ form a discrete-time Markov chain distributed as a Hall-Littlewood process, with the number of matrices in the product pla…
