相关论文: Uniform distribution in nilmanifolds along functio…

We study the equidistribution of orbits of the form $b_1^{a_1(n)}... b_k^{a_k(n)}\Gamma$ in a nilmanifold $X$, where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under…

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of…

We study equidistribution properties of nil-orbits $(b^nx)_{n\in\N}$ when the parameter $n$ is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if $X=G/\Gamma$ is a nilmanifold,…

Consider $G=\SL_{ d }(\mathbb R)$ and $ \Gamma=\SL_{ d }(\mathbb Z)$. It was recently shown by the second-named author \cite{s} that for some diagonal subgroups $\{g_t\}\subset G$ and unipotent subgroups $U\subset G$, $g_t$-trajectories of…

We study quantitative equidistribution in law of affine random walks on nilmanifolds, motivated by a result of Bourgain, Furman, Mozes and the third named author on the torus. Under certain assumptions, we show that a failure to having fast…

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $\Gamma$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\R}=(u_x)_{x\in\R}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially…

This paper is the first part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey…

We prove that for a minimal rotation T on a 2-step nilmanifold and any measure mu, the push-forward T^n(mu) of mu under T^n tends toward Haar measure if and only if mu projects to Haar measure on the maximal torus factor. For an arbitrary…

The main goal of this paper is to obtain optimal estimates on the speed of equidistribution of nilflows on higher step nilmanifolds. Under a Diophantine condition on the frequencies of the toral projection of the flow, we prove that for…

The main results of this paper are to prove bounds for ergodic averages for nilflows on general higher step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become…

Let $H$ be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let $f \in H$ be a subpolynomial function. Let $\mathcal{P} = \{2, 3, 5, 7, \dots \}$ be the (naturally…

This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In…

Given a finite group $\Gamma$, we prove results on the distribution of the prime-to-$q|\Gamma|$ part of fundamental groups of $\Gamma$-covers of the projective line $\mathbb P^1_{\mathbb F_q}$ over a finite field $\mathbb F_q$ as…

Let ${\cal H}=(q_1, \ldots q_r)$ be a finite set of coprime integers and let $n_1, n_2, \ldots$ denote the multiplicative semigroup generated by $\cal H$ and arranged in increasing order. The distribution of such sequences has been studied…

We prove quantitative equidistribution results for actions of Abelian subgroups of the $2g+1$ dimensional Heisenberg group acting on compact $2g+1$-dimensional homogeneous nilmanifolds. The results are based on the study of the…

Gaussian distributions can be generalized from Euclidean space to a wide class of Riemannian manifolds. Gaussian distributions on manifolds are harder to make use of in applications since the normalisation factors, which we will refer to as…

We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain…

For a wide range of functions $W\colon\mathbb{N}\to\mathbb{N}$, we establish a general result for estimating weighted averages of the form \[ \mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)),…

Let $(X,\mu)$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1, \dotsc, T_m$. We prove that the ergodic averages \[ A_{N;X}^{P_1, \dotsc, P_m}f = \frac{1}{N} \sum_{n=1}^N…

Let $G=SL_2(\mathbb R)^d$ and $\Gamma=\Gamma_0^d$ with $\Gamma_0$ a lattice in $SL_2(\mathbb R)$. Let $S$ be any "curved" submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an $\mathbb R$-diagonalizable…