Multiple common expansions in non-integer bases
Tóm tắt
We investigate the existence of simultaneous representations of real numbers x in bases 1 < q1 < … < qr, r ≥ 2, with a finite digit set A ⊂ ℝ. We prove that if A contains both positive and negative digits, then each real number has infinitely many common expansions. In general the bases depend on x. If A contains the digits −1, 0, 1, then there exist two non-empty open intervals I, J such that for any fixed q1 ∈ I each x ∈ J has common expansions for some bases q1 < … < qr.
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