[Paper Review] Uniform distribution of subpolynomial functions along primes and applications

Vitaly Bergelson, Grigori Kolesnik+1 more|arXiv (Cornell University)|Mar 17, 2015

Analytic Number Theory Research9 references3 citations

TL;DR

This paper establishes that a subpolynomial function $ f \in H $, where $ H $ is a Hardy field, is uniformly distributed modulo 1 along the primes if and only if its values at primes are uniformly distributed modulo 1. This equivalence enables significant generalizations of ergodic and combinatorial results, including new density and recurrence theorems for polynomial and subpolynomial sequences along primes, extending prior work in the field of uniform distribution and recurrence in dynamical systems.

ABSTRACT

Let $H$ be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let $f \in H$ be a subpolynomial function. Let $\mathcal{P} = \{2, 3, 5, 7, \dots \}$ be the (naturally ordered) set of primes. We show that $(f(n))_{n \in \mathbb{N}}$ is uniformly distributed mod 1 if and only if $(f(p))_{p \in \mathcal{P}}$ is uniformly distributed mod 1. This result is then utilized to derive various ergodic and combinatorial statements which significantly generalize the results obtained in [BKMST].

Motivation & Objective

• To establish a fundamental equivalence between uniform distribution modulo 1 of a subpolynomial function $ f \in H $ on $ \mathbb{N} $ and its restriction to the primes $ \mathcal{P} $.
• To generalize previous ergodic and combinatorial results on recurrence and density involving primes and polynomial-like sequences.
• To extend the applicability of Furstenberg’s correspondence principle to subpolynomial and polynomial sequences evaluated at shifted primes.
• To prove that the set of differences $ \{(p-1)^{\alpha_i}, [(p-1)^{\beta_j}] \} $ has positive lower relative density in sets of positive upper Banach density.
• To demonstrate that the sequence $ (f(p))_{p \in \mathcal{P}} $ is uniformly distributed mod 1 for subpolynomial $ f \in H $, under mild conditions on the function's growth and irrationality.

Proposed method

• Use of the Hardy field framework to analyze subpolynomial functions closed under differentiation and with controlled growth.
• Application of Weyl's criterion and exponential sum estimates to establish uniform distribution modulo 1 along primes.
• Reduction of the problem to equidistribution of linear and polynomial phases along primes via the circle method and Bombieri–Vinogradov-type estimates.
• Employment of spectral decomposition in Hilbert spaces into rational and totally weakly mixing components to analyze ergodic averages.
• Use of the correspondence principle to translate ergodic results into combinatorial statements about sets of positive upper Banach density.
• Construction of linear maps $ L $ to embed polynomial and subpolynomial sequences into higher-dimensional configurations for density analysis.

Experimental results

Research questions

• RQ1 Under what conditions is the sequence $ (f(p))_{p \in \mathcal{P}} $ uniformly distributed modulo 1 for a subpolynomial function $ f \in H $?
• RQ2 When does the uniform distribution of $ (f(n))_{n \in \mathbb{N}} $ mod 1 imply the same for $ (f(p))_{p \in \mathcal{P}} $?
• RQ3 Can ergodic averages along sequences of the form $ ((p-h)^{\alpha_i}, [(p-h)^{\beta_j}]) $ be shown to converge strongly in Hilbert space?
• RQ4 What is the relative density of the set $ D_h = \{ ((p-h)^{\alpha_i}, [(p-h)^{\beta_j}]) \} $ in sets of positive upper Banach density?
• RQ5 How can the correspondence principle be adapted to sequences involving both integer and fractional parts of subpolynomial functions along primes?

Key findings

• The sequence $ (f(p))_{p \in \mathcal{P}} $ is uniformly distributed modulo 1 if and only if $ (f(n))_{n \in \mathbb{N}} $ is, for any subpolynomial $ f \in H $.
• For any measure-preserving $ \mathbb{Z}^{k+l} $-action and $ \mu(A) > 0 $, the set $ \{ \mathbb{d} \in D_h : \mu(A \cap T^{-\mathbb{d}}A) \geq \mu^2(A) - \epsilon \} $ has positive lower relative density in $ D_h $ for $ h = \pm 1 $.
• For any set $ E \subset \mathbb{Z}^{k+l} $ with $ d^*(E) > 0 $, the lower density of $ \{ p \leq N : \text{the vector} \, ((p\pm1)^{\alpha_i}, [(p\pm1)^{\beta_j}]) \in E - E \} $ is bounded below by a positive constant depending only on $ d^*(E) $.
• The limit inferior of the normalized count of such primes satisfies $ \liminf_{N \to \infty} \frac{1}{\pi(N)} \left| \{ p \leq N : \cdots \in E - E \} \right| \geq d^*(E)^2 $.
• The result holds even when the exponents $ \alpha_i $ are positive integers and $ \beta_j $ are positive non-integers, extending classical results on polynomial sequences along primes.
• The method applies to linear combinations of subpolynomial functions $ \xi_j(p) $, provided their rational linear combinations remain in the Hardy field $ H $.