Finiteness of integral values for the ratio of two linear recurrences. (English) Zbl 1026.11021

Some years ago, the reviewer showed [C. R. Acad. Sci., Paris, Sér. I 306, 97-102 (1998; Zbl 0635.10007); see also R. Rumely, Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 349-382, 383-409 (1988; Zbl 0661.10017)] that if every term of the term-by-term quotient of two linear recurrence sequences is an integer then that ‘Hadamard quotient’ must itself be a linear recurrence sequence. Here the authors prove a best possible such result, thus with the minimal hypothesis that just infinitely many of those quotients be integral.
The core idea of the argument is quite different from that of the reviewer. The \(p\)-adic subspace theorem of Schmidt and Schlickewei is applied to the difference of the quotient and an appropriate approximation thereto. That has two advantages. First, one needs the integrality data only for infinitely many terms rather than for all terms and, second, the argument in the general case is not of a materially different nature from that of the ‘dominant root’ case.
The authors give a very clear explanation of their powerful arguments and of the manifold implications of their result. There is also useful discussion of related quantitative considerations.


11B37 Recurrences
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