相关论文: Geometric and o-minimal Littlewood-Offord problems

In this article, we studied the inverse Erd\H{o}s-Heilbronn problem with the restricted sumset from two components $A$ and $B$ that are not necessarily the same. We give a completely elementary proof for the problem in $\mathbb{Z}$ and some…

We investigate the relations between spinors and null vectors in Clifford algebra with particular emphasis on the conditions that a spinor must satisfy to be simple (also: pure). In particular we prove: i) a new property for null vectors:…

We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)} \sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence…

In this paper, we give a uniqueness theorem for the Dirichlet problem of minimal maps into general Riemannian manifolds with non-positive sectional curvature, improving Theorem 5.2 of Lee-Ooi-Tsui's paper published in J. Geom. Anal.. The…

In this paper we study the Mixed Littlewood Conjecture with pseudo-absolute values. We show that if p is a prime and D is a pseudo-absolute value sequence satisfying mild conditions then then the infimum over natural numbers n of the…

In this paper, we establish an improved version of a saddle point theorem ([4]) removing a weak lower semicontinuity assumption at all. We then revisit some of the applications of that theorem in the light of such an improvement. For…

These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…

The basic representation of $\A$ is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.

The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of results of Arak (1980) are…

The Erd\H{o}s distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is Erd\H{o}s' distinct angle problem, the problem of finding the minimum number of distinct angles between $n$ non-collinear…

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erd\H{o}s asked whether this implies…

For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$. This conjecture follows from the generalised Riemann hypothesis, and is known to hold…

We prove a Kazhdan-Margulis-Zassenhaus lemma for Hilbert geometries. More precisely, in every dimension $n$ there exists a constant $\varepsilon_n > 0$ such that, for any properly open convex set $\O$ and any point $x \in \O$, any discrete…

A set of integers $A$ is non-averaging if there is no element $a$ in $A$ which can be written as an average of a subset of $A$ not containing $a$. We show that the largest non-averaging subset of $\{1, \ldots, n\}$ has size $n^{1/4+o(1)}$,…

We introduce and discuss an Erd\H{o}s-Ko-Rado basis for the underlying vector space of a Leonard system $\Phi = (A; A^*; \{E_i\}_{i=0}^d ; \{E_i^* \}_{i=0}^d)$ that satisfies a mild condition on the eigenvalues of $A$ and $A^*$. We describe…

We show that for a minimal system $(X,T)$, the set of saturated points along cubes with respect to its maximal $\infty$-step pro-nilfactor $X_\infty$ has a full measure. As an application, it is shown that if a minimal system $(X,T)$ has no…

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We verify the so-called Zilber-Pink…

Two comparison theorems are established for discrete eigenvalues of the Klein-Gordon equation with an attractive central vector potential in d >= 1 dimensions. (I) If \psi_1 and \psi_2 are node-free ground states corresponding to positive…

The eigenvalues of a self-adjoint nxn matrix A can be put into a decreasing sequence $\lambda=(\lambda_1,...,\lambda_n)$, with repetitions according to multiplicity, and the diagonal of A is a point of $R^n$ that bears some relation to…

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…