相关论文: New Kronecker-Weyl type equidistribution results a…

We establish various analogs of the Kronecker-Weyl equidistribution theorem that can be considered higher-dimensional versions of results established in our earlier investigation of the discrete 2-circle problem studied in 1969 by Veech.…

It is well known that (i) for every irrational number $\alpha$ the Kronecker sequence $m\alpha$ ($m=1,...,M$) is equidistributed modulo one in the limit $M\to\infty$, and (ii) closed horocycles of length $\ell$ become equidistributed in the…

We study the relationship between the discrete and the continuous versions of the Kronecker--Weyl equidistribution theorem, as well as their possible extension to manifolds in higher dimensions. We also investigate a way to deduce in some…

Given any rectangular polyhedron 3-manifold $P$ tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in…

The parity of the partition function $p(n)$ remains strikingly mysterious. Beyond a handful of fragmentary results, essentially nothing is known about the distribution of parity. We prove a uniform result on quadratic progressions. If…

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n \alpha)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $\alpha$. Another…

Measure rigidity is a branch of ergodic theory that has recently contributed to the solution of some fundamental problems in number theory and mathematical physics. Examples are proofs of quantitative versions of the Oppenheim conjecture,…

For irrational $\alpha$, $\{n\alpha\}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences…

Consider the sequence of continued fraction convergents $p_n/q_n$ to a random irrational number. We study the distribution of the sequences $p_n \pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$, and more generally, the distribution of…

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and…

The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss…

For the generalized Dedekind sums s_{ij}(p,q) defined in association with the x^{i}y^{j}-coefficient of the Todd power series of the lattice cone in R^2 generated by (1,0) and (q,p), we associate an exponential sum. We obtain this…

A classical theorem of Weyl states that any polynomial with an irrational coefficient other than the constant term is uniformly distributed mod 1. We prove a new function field analogue of this statement, confirming a conjecture of L\^{e},…

An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} \alpha \right)_{n\geq 1}$ is uniformly distributed modulo one for…

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special…

We show that given a closed $n$-manifold $M$, for a generic set of Riemannian metrics $g$ on $M$ there exists a sequence of closed geodesics that are equidistributed in $M$ if $n=2$; and an equidistributed sequence of embedded stationary…

We investigate actions of locally compact Abelian (LCA) groups on the torus $\mathbb{T}^n$, motivated by their close connection with Diophantine approximation. While Kronecker's theorem yields a classical density result, we prove a stronger…

We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of $m$-regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for…

H. Weyl proved in \cite{Weyl} that integer evaluations of polynomials are equidistributed mod 1 whenever at least one of the non-free coefficients is irrational. We use Weyl's result to prove a higher dimensional analogue of this fact.…

We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2(…