Khintchine's Theorem with random fractions

TL;DR

This paper extends Khintchine's Theorem to random rational approximations, analyzing how the removal of monotonicity affects approximation sets and exploring connections to the Duffin-Schaeffer and Catlin conjectures.

Contribution

It introduces probabilistic versions of Khintchine's Theorem for random fractions and investigates the impact of growth conditions on the necessity of monotonicity.

Findings

Monotonicity can be removed if the number of fractions grows slowly.

Fast growth of available fractions makes monotonicity essential.

Connections established between random Khintchine, Duffin-Schaeffer, and Catlin conjectures.

Abstract

We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin-Schaeffer Conjecture (1941), and are likely to be more accessible. We point out that the direct random analogue of the Duffin-Schaeffer Conjecture, like the Duffin-Schaeffer Conjecture itself, implies Catlin's Conjecture (1976). It is not obvious whether the Duffin-Schaeffer Conjecture and its random version imply one another, and it is not known whether Catlin's Conjecture implies either of them. The question of whether Catlin implies Duffin-Schaeffer has been unsettled for decades.