Property tau in positive characteristic

Effective equidistribution and property tau

We prove a quantitative equidistribution statement for adelic homogeneous subsets whose stabilizer is maximal and semisimple. Fixing the ambient space, the statement is uniform in all parameters. We explain how this implies certain equidistribution theorems which, even in a qualitative form, are not accessible to measure-classification theorems. As another application, we describe another proof of property tau for arithmetic groups.

Characterization algorithms for shift radix systems with finiteness property

For a natural number d and a d-dimensional real vector r let Tau(r) denote the (d-dimensional) shift radix system associated with r. Tau(r) is said to have the finiteness property iff all orbits of Tau(r) end up in the zero vector; the set of all corresponding parameters r is denoted by DN(d), whereas D(d) consists of those parameters r for which all orbits are eventually periodic. DN(d) has a very complicated structure even for d=2. In the present paper two algorithms are presented which allow the characterization of the intersection of DN(d) and any closed convex hull of finitely many interi…

On the $P_1$ property of sequences of positive integers

It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence. Certainly, there is a proof based on the Chinese Remainder Theorem. In this paper we give proofs of two analytic criteria revealing this property of sequences.

Liminf Sets in Simultaneous Diophantine Approximation

Let Q be an infinite set of positive integers. Denote by W(Q) the set of n-tuples of real numbers simultaneously tau-well approximable by infinitely many rationals with denominators in Q but only by finitely many rationals with denominators in the complement of Q. The Hausdorff dimension of the liminf set W(Q) is computed when tau > 2 + 1/n. A p-adic analogue of the problem is also studied.

Odd values of the Ramanujan tau function

We prove a number of results regarding odd values of the Ramanujan $τ$-function. For example, we prove the existence of an effectively computable positive constant $κ$ such that if $τ(n)$ is odd and $n \ge 25$ then either \[ P(τ(n)) \; > \; κ\cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p \mid n$ with $τ(p)=0$. Here $P(m)$ denotes the largest prime factor of $m$. We also solve the equation $τ(n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$ and the equations $τ(n)=\pm q^b$ where $3\le q < 100$ is prime and the exponents are arbitrary nonnegative integers. We make use of…

关于正则分拆函数、广义 Tau 函数与分拆函数加权复合和的同余关系

Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums

本文将广义 τ_k(n) 表示为二项式系数加权分拆和，从而建立其与 t-正则分拆函数 R_t(n) 及加权复合和之间的同余恒等式。