An irrationality measure for Liouville numbers and conditional measures for Euler's constant

Abstract

The irrationality exponent $\mu(t)$ of an irrational number t, defined using the irrationality measure $1/q^\mu$, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure $1/\beta^q$, we define the "irrationality base" $\beta(t)$, which distinguishes among Liouville numbers and is 1 for non-Liouville numbers. We give some properties and examples. Assuming a condition on certain linear forms in logarithms, for which we present numerical evidence supplied by P. Sebah, we prove an upper bound on the irrationality base of Euler's constant, $\gamma$. If $\gamma$ is irrational and the condition turns out to be false in a certain strong sense, we prove an upper bound on $\mu(\gamma)$.