Squaring the terms of an $\ell^{th}$ order linear recurrence

递归与关于最小部分函数的同余关系

Recurrence and congruences for the smallest parts function

本文基于Hecke迹研究最小部分函数的递归公式，并推导出同余表达式，同时证明了该函数的非同余性质。

Quantitative multiple recurrence for two and three transformations

We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2)$ with two commuting transformations such that for every $0<\ell< 4$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<μ(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},μ,T_1,T_2, T_{3})$ with three commuting transformations such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$μ(A\cap T_{1}…

Simultaneous Polynomial Recurrence

Let $A\subseteq\{1,...,N\}$ and $P_1,...,P_\ell\in\Z[n]$ with $P_i(0)=0$ and $°P_i=k$ for every $1\leq i\leq\ell$. We show, using Fourier analytic techniques, that for every $\VE>0$, there necessarily exists $n\in\N$ such that \[\frac{|A\cap (A+P_i(n))|}{N}>(\frac{|A|}{N})^2-\VE\] holds simultaneously for $1\leq i\leq \ell$ (in other words all of the polynomial shifts of the set $A$ intersect $A$ "$\VE$-optimally"), as long as $N\geq N_1(\VE,P_1,...,P_\ell)$. The quantitative bounds obtained for $N_1$ are explicit but poor; we establish that $N_1$ may be taken to be a constant (depending only …

Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K

We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of higher order q-recurrence equations with rational coefficients. We extend a method for finding a bound on the maximal power of t in the denominator of arbitrary rational solutions y(t) as well as a method for bounding the degree of polynomial solutions from the scalar case to the systems case. The approach is direct and does not rely on uncoupling or reduction to a first order system. Unlike in the scalar case this usually requires an initial transformation of the system.

Multiple recurrence and convergence for Hardy sequences of polynomial growth

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for "typical" choices of Hardy field functions $a(t)$ with polynomial growth, the averages $\frac{1}{N}\sum_{n=1}^N f_1(T^{[a(n)]}x)\cdot...\cdot f_\ell(T^{\ell [a(n)]}x)$ converge in the mean and we determine their limit. For example, this is the case if $a(t)=t^{3/2}, t\log{t},$ or $t^2+(\log{t})^2$. Furthermore, if ${a_1(t),...,a_\ell(t)}$ is a "typical" family of logarithmico-exponential functions of polynomial growth, …

混合多重卷积递推的增长性

Growth of recurrences with mixed multifold convolutions

本文研究一类推广 Catalan 数与 Schröder 数的混合多重卷积递推序列，证明其增长速率可任意精度计算，并给出上下界比值渐近收敛到 1 的显式不等式。