Developments in Non-Integer Bases

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We prove various theorems concerning the developments in non-integer bases. We mention two of them here, which answer some questions formulated several years ago. First fix a real number q > 1 and consider the increasing sequence 0 = yo < y1 < y2 < … of those real numbers y which have at least one representation of the form y = ε0 + ε1q + … + εnqn with some integer n ≧ 0 and coefficients ε, Ε {0, 1}. Then the difference sequence yk+1 − yk tends to 0 for all q, sufficiently close to 1.

Secondly, for each q, sufficiently close to 1, there exists a sequence (εi) of zeroes and ones, satisfying ∑i=1∞εiq−i = 1 and containing all possible finite variations of the digits 0 and 1.

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