ON p-ADIC INTERPOLATION IN TWO OF MAHLER’S PROBLEMS
Motivated by the p-adic approach in two of Mahler’s problems, we obtain some results on p-adic analytic interpolation of sequences of integers $(u_n)_{n\geq 0}$ . We show that if $(u_n)_{n\geq 0}$ is a sequence of integers with $u_n = O(n)$ which can be p-adically interpolated by an analytic functio...
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Veröffentlicht in: | Bulletin of the Australian Mathematical Society 2023-08, Vol.108 (1), p.69-80 |
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Sprache: | eng |
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Zusammenfassung: | Motivated by the p-adic approach in two of Mahler’s problems, we obtain some results on p-adic analytic interpolation of sequences of integers
$(u_n)_{n\geq 0}$
. We show that if
$(u_n)_{n\geq 0}$
is a sequence of integers with
$u_n = O(n)$
which can be p-adically interpolated by an analytic function
$f:\mathbb {Z}_p\rightarrow \mathbb {Q}_p$
, then
$f(x)$
is a polynomial function of degree at most one. The case
$u_n=O(n^d)$
with
$d>1$
is also considered with additional conditions. Moreover, if X and Y are subsets of
$\mathbb {Z}$
dense in
$\mathbb {Z}_p$
, we prove that there are uncountably many p-adic analytic injective functions
$f:\mathbb {Z}_p\to \mathbb {Q}_p$
, with rational coefficients, such that
$f(X)=Y$
. |
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ISSN: | 0004-9727 1755-1633 |
DOI: | 10.1017/S0004972722000946 |