Abstract
We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.
Highlights
Expansions of reals numbers in non-integer bases are studied since the pioneering works of Renyi in the end of the 1950s and Parry in the 1960s, see [11,12,13]
In the 1990s, a group of Hungarian mathematics led by Paul Erdos revived this file of research, see [3,4,5]
We introduce non-uniform expansions of real numbers, which may be viewed as expansions with respect to two non-integer bases
Summary
Expansions of reals numbers in non-integer bases are studied since the pioneering works of Renyi in the end of the 1950s and Parry in the 1960s, see [11,12,13]. In the 1990s, a group of Hungarian mathematics led by Paul Erdos revived this file of research, see [3,4,5] Beside other results, they proved that each x ∈ (0, 1/(1 − q)) has a continuum of expansions of the form. We prove a theorem on the existence of a continuum of nonuniform expansions of real numbers, which is similar to the results in the uniform case we mentioned above.
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