Irrationality exponent and rational approximations with prescribed growth

TL;DR

This paper investigates the existence of sequences of linear forms in 1 and an irrational number with prescribed growth and decay rates, establishing conditions related to the irrationality exponent that are both necessary and sufficient.

Contribution

It proves that the necessary condition involving the irrationality exponent is also sufficient for the existence of such sequences with arbitrary growth and decay rates.

Findings

Established a precise condition involving the irrationality exponent for the existence of linear forms with prescribed growth and decay.

Proved the sufficiency of this condition even for arbitrary rates, extending previous understanding.

Discussed potential multivariate generalizations related to linear independence criteria.

Abstract

Let $\xi$ be a real irrational number. We are interested in sequences of linear forms in 1 and $\xi$, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of $\xi$. We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin's new proof of Nesterenko's linear independence criterion.