Hardy域中函数沿加权均匀分布的一个准则
A criterion for weighted uniform distribution along functions from a Hardy field
AI总结：
本文利用可和性理论方法，将Boshernitzan关于Hardy域函数模1均匀分布的经典准则推广到加权平均情形，并应用于函数f(x)=x^{3/2}得到区间内点的存在性。
摘要：
Boshernitzan的一个经典定理指出：若$f$属于Hardy域且对某个$\ell\in \mathbb{N}$满足$|f(x)|\prec x^{\ell}$，则序列$(f(n))_{n\in \mathbb{N}}$模1均匀分布当且仅当对所有$p(x)\in \mathbb{Q}[x]$有$\lim_{x\to\infty}\frac{|f(x)-p(x)|}{\log(x)} = \infty$。我们利用可和性理论方法给出了该结果的一个新证明，并通过获得$f$关于一大类加权平均模1均匀分布的充要条件，推广了Boshernitzan准则。作为我们结果的一个应用，我们证明：对函数$f(x) = x^{3/2}$，任意$(a,b)\subset [0,1]$和所有充分大的$N\in\mathbb{N}$，存在$n\in [N-N^{\frac{1}{4}},N]$使得$f(n)\mod 1\in (a,b)$。
英文摘要：
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 $f(n)\mod 1\in (a,b)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
