Overview
This research investigates the weighted uniform distribution of subpolynomial functions. Specifically, it addresses sequences of the form $(u(n))_{n\in\mathbb{N}}$ and $(u(p_n))_{n\in\mathbb{N}}$, where $u(x)$ denotes a subpolynomial function originating from a Hardy field, and $p_n$ represents the $n$-th prime number.
Research Context
The work directly builds upon and extends a prior main result described in reference [4]. The extension involves incorporating a weighted setting into the analysis of uniform distribution. This contextualization highlights the progressive nature of the research within its mathematical domain, specifically in relation to existing foundational work.
Findings
The study establishes criteria identified as both necessary and sufficient for the weighted uniform distribution of the aforementioned sequences. These conditions apply to subpolynomial functions $u(x)$ that are elements of a Hardy field. The specific sequences under examination are $(u(n))_{n\in\mathbb{N}}$, which relates to the natural numbers, and $(u(p_n))_{n\in\mathbb{N}}$, which specifically addresses the distribution along prime numbers.
Why This Matters
The established necessary and sufficient conditions for weighted uniform distribution contribute to the development of uniform distribution theory. The extension of the main result from reference [4] to a weighted setting yields new applications. These applications are explicitly stated to encompass uniform distribution theory itself, as well as ergodic theory and additive combinatorics, suggesting a multi-disciplinary impact within pure mathematics.