On the distribution of polynomial Farey points and Chebyshev's bias phenomenon

TL;DR

This paper investigates the distribution and pair correlation of polynomial Farey fractions, establishing uniform distribution, explicit pair correlation formulas, and Chebyshev bias phenomena, with results varying between Poissonian and non-Poissonian behaviors.

Contribution

It provides the first detailed analysis of the distribution, pair correlation, and Chebyshev bias for polynomial Farey fractions, including explicit formulas and distribution properties.

Findings

Polynomial Farey fractions are uniformly distributed modulo one.

Explicit non-Poissonian pair correlation for P(x)=x(x+1).

Poissonian pair correlation when denominators are restricted to primes.

Abstract

We study two types of problems for polynomial Farey fractions. For a positive integer $Q$, and polynomial $P(x)\in\mathbb{Z}[X]$ with $P(0)=0$, we define polynomial Farey fractions as \[\mathcal{F}_{Q,P}:=\left\{\frac{a}{q}: 1\leq a\leq q\leq Q,\ \gcd (P(a),q)=1\right\}.\] The classical Farey fractions are obtained by considering $P(x)=x$. In this article, we determine the global and local distribution of the sequence of polynomial Farey fractions via discrepancy and pair correlation measure, respectively. In particular, we establish that the sequence of polynomial Farey fractions is uniformly distributed modulo one and show that the limit superior of the pair correlation measure of $(\mathcal{F}_{Q,P})_{Q\ge1}$ is bounded. For the specific polynomial $P(x)=x(x+1)$, we show the existence of the limiting pair correlation measure of $(\mathcal{F}_{Q,P})_{Q\ge1}$ and also provide an explicit formula for the pair correlation function which is non-Poissonian. Further, restricting the polynomial Farey denominators to certain subsets of primes, we explicitly find the pair correlation measure and show it to be Poissonian. Finally, we study Chebyshev's bias type of questions for the classical and polynomial Farey denominators along arithmetic progressions and obtain an $\Omega$-result for the error term of its counting function.