相关论文: Resolution of the quadratic Littlewood--Offord pro…

Consider a degree-$d$ polynomial $f(\xi_1,\dots,\xi_n)$ of independent Rademacher random variables $\xi_1,\dots,\xi_n$. To what extent can $f(\xi_1,\dots,\xi_n)$ concentrate on a single point? This is the so-called polynomial…

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ 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,…

The classical Erd\H{o}s-Littlewood-Offord problem concerns the random variable $X = a_1 \xi_1 + \dots + a_n \xi_n$, where $a_i \in \mathbb{R} \setminus \{0\}$ are fixed and $\xi_i \sim \text{Ber}(1/2)$ are independent. The…

Consider the sum $X(\xi)=\sum_{i=1}^n a_i\xi_i$, where $a=(a_i)_{i=1}^n$ is a sequence of non-zero reals and $\xi=(\xi_i)_{i=1}^n$ is a sequence of i.i.d. Rademacher random variables (that is, $\Pr[\xi_i=1]=\Pr[\xi_i=-1]=1/2$). The…

Consider nonzero vectors $a_{1},\dots,a_{n}\in\mathbb{C}^{k}$, independent Rademacher random variables $\xi_{1},\dots,\xi_{n}$, and a set $S\subseteq\mathbb{C}^{k}$. What upper bounds can we prove on the probability that the random sum…

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…

We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which…

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…

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…

We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. We discuss…

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums $\sum_{k=1}^{n} X_k a_k$ with respect to the arithmetic structure of…

For a random variable $X$ define $Q(X) = \sup_{x \in \mathbb{R}} \mathbb{P}(X=x)$. Let $X_1, \dots, X_n$ be independent integer random variables. Suppose $Q(X_i) \le \alpha_i \in (0,1]$ for each $i \in \{1, \dots, n\}$. Ju\v{s}kevi\v{c}ius…

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}…

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums $\sum_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients…

Let $\epsilon_1, \dotsc, \epsilon_n$ 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…

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…

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.,…

We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some "Poisson-type"…

We give a new bound on the probability that the random sum $\xi_1 v_1 +...+ \xi_n v_n$ belongs to a ball of fixed radius, where the $\xi_i$ are iid Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an application, we prove…

The classical Erd\H{o}s-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 $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have…