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

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The irrationality exponent μ(t) of an irrational number t, defined using the irrationality measure 1/qμ, distinguishes among non-Liouville numbers and is infinite for Liouville numbers. Using the irrationality measure 1/βq, we define the "irrationality base" β(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, γ. If γ is irrational and the condition turns out to be false in a certain strong sense, we prove an upper bound on μ(γ).