H. Weyl在文献[Weyl]中证明了：只要多项式的非自由系数中至少有一个是无理数，该多项式在整数点上的取值模1就是等分布的。我们利用Weyl的结果证明这一事实的一个高维类比。具体而言，我们证明：只要非自由系数中至少有一个是无理数，多项式在格点上的取值模1就是等分布的。这一结果加强了Arhipov-Karacuba-Čubarikov在文献[PolWeyl]中的主要结果。我们将该类比作为一个定理的推论加以证明，该定理保证：对于所有满足一定导数限制条件的函数，其在格点上的取值模1都是等分布的。我们证明的另一个推论是：对于$p\in(1,\infty)$，整数向量的$\ell^p$范数模1是等分布的。
英文摘要：
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. Namely, we prove that evaluations of polynomials on lattice points are equidistributed mod 1 whenever at least one of the non-free coefficients is irrational. This result strengths the main result of Arhipov-Karacuba-Čubarikov in \cite{PolWeyl}. We prove this analogue as a Corollary of a Theorem that guarantees equidistribution of lattice evaluations mod 1 for all functions which satisfy some restrains on their derivatives. Another Corollary we prove is that for $p\in(1,\infty)$ the $\ell^p$ norms of integer vectors are equidistributed mod 1.
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 $P$.
In this note, we improve earlier results towards the Bruinier-Kohnen sign equidistribution conjecture for half-integral weight modular eigenforms in terms of natural density by using a consequence of Halász' Theorem. Moreover, applying a result of Serre we remove all unproved assumptions.
In this expository note, we present an approach to the generalization of Serre of the Sato-Tate Conjecture. Most of its content is taken from Serre's original references. However, we provide a few new examples and supply references to recent progress developed in the area.
The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed (mod 1) sequences, in the "metric" (meaning almost sure stochastic…
A deterministic sequence of real numbers in the unit interval is called \emph{equidistributed} if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a sequence is called Poissonian if the number of pairs $x_k,x_l \in (x_n)_{1 \leq n \leq N}$ which are within distance $s/N$ of each other is asymptotically $\sim 2sN$. A randomly generated sequence has both of these properties, almost surely. There seems to be a vague sense that having Poissonian pair correlations is a "finer" property than being equidistribu…
In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretèche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher, and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.
In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in $\mathbb{Z}^d$. In particular, we prove that if $Γ\leq \operatorname{GL}_d(\mathbb{Z})$ is finitely generated by unipotents and acts irreducibly on $\mathbb{R}^d$, then for any set $B \subset \mathbb{Z}^d$ of positive density, there exists $k \geq 1$ such that for any $v \in k \mathbb{Z}^d$ one can find $γ\in Γ$ with $γv \in B - B$. Our method does not require the linearity of the action, and we prove a twisted recurrence for semigroups of maps from $\mathbb{Z}^d$ to $\mathb…
