3
votes
1answer
209 views

Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*} f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...
3
votes
0answers
92 views

Equidistribution of double coset

Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the ...
12
votes
1answer
302 views

Is $\lfloor \log(n!)\rfloor \alpha$ equidistributed on the unit circle?

In this question $\lfloor a\rfloor$ means the greatest integer not exceeding $a$. Using van der Corput's inequalities one is able to show that $\log(n!)\alpha$ is equidistributed on the unit circle ...
4
votes
1answer
128 views

Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
6
votes
2answers
242 views

Minimal period of arithmetic progressions occurring in sets of positive density.

Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ ...
4
votes
3answers
347 views

Equidistibution of horocycles through Hecke eigenvalues of Maass cusp forms

At the end of this very nice post: http://blogs.ethz.ch/kowalski/2012/05/21/who-needled-buffon/ E. Kowalski talks about the equidistribution of the points $\frac{j+i}{N}$ when $j=1,\dots,N$ and $N$ ...
1
vote
0answers
283 views

What is the mean-value of a particular exponential sum related to the non-trivial zeros of Riemann's zeta function?

This question arose from an earlier one and the MO user's useful answers there: What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (which is not a ...
7
votes
2answers
419 views

A non-standard ergodic limit

Suppose $T$ is an ergodic measure-preserving transformation on a measure space $(X,\Sigma,\mu)$, and $f\in L^1(\mu)$. Does the limit $\lim_{X\to\infty} \pi(X)^{-1}\sum_{p\leq X} f(T^{p}x)$ exist ...
7
votes
2answers
235 views

a.e. convergence of the powers of an operator built from rotations

Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by $$T(f)(x)=1/2(f(x+a)+f(x+b))$$ For which values of $a,b$ do we have almost ...