Approximation by random fractions

TL;DR

This paper investigates the approximation of real numbers in the unit interval by randomly chosen rational fractions, establishing conditions under which classical approximation theorems hold without monotonicity assumptions.

Contribution

It extends previous results by showing that Khintchine-like theorems can be valid without monotonicity if probabilities decay sufficiently fast with denominator size.

Findings

Khintchine-like results hold under fast decay of probabilities

Monotonicity assumption can be removed with certain decay conditions

Results depend on the decay rate of the probability model

Abstract

We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the question of when the monotonicity assumption can be removed. Informally speaking, we show that if the probabilities in our model decay sufficiently fast as the denominator increases, then a Khintchine-like statement holds without a monotonicity assumption. Although our rate of decay of probabilities is unlikely to be optimal, it is known that such a result would not hold if the probabilities did not decay at all.