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ON p-ADIC INTERPOLATION IN TWO OF MAHLER’S PROBLEMS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 1 / August 2023
- Published online by Cambridge University Press:
- 22 September 2022, pp. 69-80
- Print publication:
- August 2023
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- Article
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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 
$f(X)=Y$.
