arXiv subjects

Michael Reilly

Publications and source records attributed to Michael Reilly.

6 recordsLinked to original sources

arXiv2026

A criterion for weighted uniform distribution along functions from a Hardy field

A classical theorem of Boshernitzan states that if $f$ is a function which belongs to a Hardy field and which satisfies $|f(x)|\prec x^{\ell}$ for some $\ell\in \mathbb{N}$, then the sequence $(f(n))_{n\in \mathbb{N}}$ is uniformly distributed modulo 1 if and only if $\lim_{x\to\infty}\frac{|f(x)-p(x)|}{\log(x)} = \infty$ for all $p(x)\in \mathbb{Q}[x]$. We provide a new proof of this result using methods from summability theory and we extend Boshernitzan's criterion by obtaining necessary and sufficient conditions for $f$ to be uniformly distributed modulo 1 with respect to a broad class of weighted averages. As an application of our results, we show that for the function $f(x) = x^{3/2}$ and for any $(a,b)\subset [0,1]$, and all sufficiently large $N\in\mathbb{N}$, there is an $n\in [N-N^{\frac{1}{4}},N]$ such that $f(n)\mod 1\in (a,b)$.

math.NT↗

arXiv2026

Weighted averages of arithmetic functions and applications to equidistribution and ergodic theory

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\[\mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)),\]where $f\colon \{1,\ldots,N\}\to\mathbb{C}$ is an arbitrary function, and $\vartheta(n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (For instance, one may take $\vartheta(n)=\Omega(n)$, where $\Omega(n)$ counts the number of prime factors of $n$ with multiplicity, or $\vartheta(n)=s_q(p_n)$, where $s_q$ is the sum-of-digits function in base $q$ and $p_n$ denotes the $n$-th prime. Additional natural examples are discussed in the paper.) Building on our main theorem, we show that if $h(n)$ is a function from a Hardy field with polynomial growth then $(h(\vartheta(n)))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) $\lim_{x\to\infty} \frac{|h(x)-p(x)|}{x \log x}=\infty$ for all $p(x)\in \mathbb{Q}[x]$; (ii) $\lim_{x\to\infty}\frac{|h(x)-p(x)|}{\sqrt{x}}=\infty$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that $\lim_{x\to\infty}\frac{|h(x)-q(x)|}{x}<\infty$. This leads to several novel applications. For example, it follows that $(\Omega(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.

math.NT↗

arXiv2026

A Composition Theorem for Binomially Weighted Averages

We study binomially weighted summation methods given by \[ (x_n)_{n\in \mathbb{N}} \mapsto \left(\sum_{k=0}^n\binom{n}{k}r^k(1-r)^{n-k}x_k\right)_{n\in \mathbb{N}} \] for $r\in (0,1)$, and their behavior under composition with summation methods of the form \[ (x_n)_{n\in \mathbb{N}} \mapsto \left(\sum_{k=0}^n\lambda_k x_{n-k}\right)_{n\in \mathbb{N}}. \] Our main result shows that if the binomially weighted averages of a sequence $(x_n)_{n\in \mathbb{N}}$ converge to a limit then the binomially weighted averages of the sequence $\left(\sum_{k=0}^n\lambda_kx_{n-k}\right)_{n\in \mathbb{N}}$ converge to the same limit whenever $(\lambda_n)_{n\in\mathbb{N}}$ is an absolutely summable sequence with $\sum_{k=0}^{\infty}\lambda_k = 1$. This result disproves a theorem appearing in the literature. Additionally, we discuss applications and extensions of our main result to compositions with weighted Ces\`aro averages.

math.GM↗

arXiv2026

Uniform Weighted Averages and a Conjecture of Bergelson, Moreira, and Richter

We confirm a conjecture posed by Bergelson, Moreira, and Richter (arXiv:1711.05729), and in particular show that for every probability measure preserving system $(X,\mathscr{B},\mu,T)$, every $k\in \mathbb{N}$, every set $A\in \mathscr{B}$ with $\mu(A)>0$, and every tempered function $f$, \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu(A\cap T^{-\lfloor{f(n)\rfloor}}A\cap T^{-\lfloor{f(n+1)\rfloor}}A\cap \cdots \cap T^{-\lfloor{f(n+k)\rfloor}}A)>0. \] This is achieved by establishing conditions on an increasing function $W:\mathbb{N}\rightarrow (0,\infty)$ such that if $(x_n)_{n\in \mathbb{N}}$ is a bounded sequence in a Banach space with \[ \lim_{W(N)-W(M)\to\infty}\frac{1}{W(N)-W(M)}\sum_{n=M}^N (W(n)-W(n-1))x_n =L \] then the limit of Ces\`aro averages of $(x_n)_{n\in \mathbb{N}}$, $\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^Nx_n$ is also equal to $L$. Furthermore, the methods we develop can be used to sharpen some of the combinatorial results obtained by Bergelson, Moreira, and Richter. For example, if $E$ is a set of positive upper density, then for any $k\in \mathbb{N}$, any $\epsilon>0$, and all sufficiently large $N\in \mathbb{N}$ there is an $n\in [N-N^{1/2+\epsilon},N]$ such that \[\{a,a+\lfloor{n^{3/2}\rfloor},a+\lfloor{(n+1)^{3/2}\rfloor},\dots ,a +\lfloor{(n+k)^{3/2}\rfloor}\}\subseteq E. \]

math.DS↗

arXiv2025

Recurrence Relations for Cosets in Free Groups

Let $F_2$ be the free group on two generators and let $H$ be a subgroup of $F_2$. We investigate a method for calculating the number of elements in a coset of $H$ that have a given length when written in reduced form. More specifically, taking $S_n\subseteq F_2$ to be the set of elements of length $n$, we show that for any coset $yH$ there always exists a recurrence relation of the form \[ |yH\cap S_n| = \sum_{i=1}^{n-1}\sum_{xH\in F_2/H}a_{i,xH}\cdot |xH\cap S_{n-i}| \] for some constants $(a_{i,xH})_{i\in \mathbb{N}, xH\in F_2/H}$, and we give an algorithm that calculates these constants. Further, we show that when $H$ has finite index and contains an element of odd length, only finitely many of the constants $a_{i,xH}$ are nonzero.

math.GR↗

arXiv2023

An Extension of Stanley's Symmetric Acyclicity Theorem to Signed Graphs

In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number of sinks. In this paper, we generalize this result to signed graphs, that is, to graphs whose edges are labeled with $+$ or $-$ and whose colorings and orientations can interact with their signs. Additionally, we introduce a non-homogeneous basis which detects the number of sinks and which not only gives a Stanley-type result for signed graphs but gives an analogous result of this form for unsigned graphs as well.

math.CO↗