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\(p\)-adic asymptotic properties of constant-recursive sequences. (English) Zbl 1356.11043

In the present article, the authors study \(p\)-adic properties of sequences of integers (or \(p\)-adic integers) satisfying a linear recurrence with constant coefficients. Such sequences cannot generally be interpolated to \(\mathbb{Z}_p\) but, as is shown in Theorem 7, every such sequence has an approximate twisted interpolation to \(\mathbb{Z}_p\) (as defined in Section 3). The authors then use this interpolation for two applications. The first is that certain subsequences of recursive sequences with constant coefficients converge \(p\)-adically. The second is that the density of the residues modulo \(p^\alpha\) attained by a constant-recursive sequence converges, as \(\alpha\to\infty\), to the Haar measure of a certain subset of \(\mathbb{Z}_p\). To illustrate these results, the authors give a twisted interpolation for the Fibonacci sequence to \(\mathbb{Z}_p\), establish some particular limits, and determine the limiting density of residues attained by the Fibonacci sequence modulo powers of 11.

MSC:

11J61 Approximation in non-Archimedean valuations
11B37 Recurrences
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References:

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