Common expansions in noninteger bases
Vilmos Komornik, Attila Pethő · Publicationes Mathematicae Debrecen · 2014
In this paper we study the existence of simultaneous representations of real numbers in bases p > q > 1 with the digit set A = {-m, . . ., 0, . . ., m}.Among other results, we prove if m = 1 and q < 2, then there is a continuum of sequencesOn the other hand, if m = 1 and q ≥ 2 + √ 2, then only the trivial sequence (c i ) = 0 ∞ satises the former equality.In case 0 ∈ A a trivial example is x = 0 with (c i ) = 0 ∞ .If the alphabet A contains no pair of digits with opposite signs, then this is the only such example.Indeed, if for instance all digits are nonnegative and