On the intersection of infinite geometric and arithmetic progressions
The question of how many numbers a given positive increasing geometric progression and a given arithmetic progression have in common is studied. When the common ratio of the geometric progression is the \(d\)th root of a rational number then the cardinality of this intersection can be any nonnegative integer, or infinity. Otherwise, the authors prove that this cardinality can be at most \(3\), except in the case when the common ratio is the \(d\)th root \(\beta^{1/d}\) of a cubic irrational \(\beta\), when they prove that it is at most \(6\). They conjecture that the bound should, in fact, also be \(3\) in this latter case. The proofs involve the study of the multiplicity of occurrence of a number \(t\in[0,1)\) as the fractional part of one of the geometric progressions being studied. In the case when the common ratio is \(\beta^{1/d}\), as above, the authors use in their proof a deep result of \textit{F. Beukers} [``The zero-multiplicity of ternary recurrences'', Compos. Math. 77, No. 2, 165--177 (1991; Zbl 0717.11012)] on the zero-multiplicity of ternary recurrence sequences. They also state, prove and use some nice elementary lemmas, which may be of interest in their own right, or useful for other applications.
scientific article; zbMATH DE number 5826902
scientific article