UNIQUE REPRESENTATIONS OF REAL NUMBERS IN NON-INTEGER BASES

By P. Glendinning, Nikita Sidorov. 2001

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Generalised golden ratios over integer alphabets (2012, background): “In [9] the authors study the case where m = 1, they show that the following theorems hold.”
Almost every number has a continuum of beta-expansions (2003, background): “We begin with an auxiliary claim that is in fact known; for example, [ 2, Theorem 2 ] yields it in even a stronger form.”
Hausdorff dimension of univoque sets and Devil's staircase (2015, background): “2 because U ′ q is countable by [15] (for M = 1) and [8], [25], [24] (for all M ≥ 1) and therefore D(q) = 0.”
Arithmetic Dynamics (2002, background): “As with [32], this is a part of the author’s credo: an elementary result deserves an elementary proof.”