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$U$-NUMBERS IN FIELDS OF FORMAL POWER SERIES OVER FINITE FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 29 July 2019, pp. 218-225
- Print publication:
- April 2020
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- Article
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In the field
$\mathbb{K}$ of formal power series over a finite field 
$K$, we consider some lacunary power series with algebraic coefficients in a finite extension of 
$K(x)$. We show that the values of these series at nonzero algebraic arguments in 
$\mathbb{K}$ are 
$U$-numbers.
A NOTE ON LACUNARY POWER SERIES WITH RATIONAL COEFFICIENTS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 11 November 2015, pp. 372-374
- Print publication:
- June 2016
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- Article
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In this note, we prove that for any
${\it\nu}>0$, there is no lacunary entire function 
$f(z)\in \mathbb{Q}[[z]]$ such that 
$f(\mathbb{Q})\subseteq \mathbb{Q}$ and 
$\text{den}f(p/q)\ll q^{{\it\nu}}$, for all sufficiently large 
$q$.
