Arithmetic properties of series of some classes

E. S. Krupitsin

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 374-382.

Résumé

The paper studies properties of Liouvillean numbers in $p$-adic, $g$-adic, polyadic domains. The canonical representation of $p$-adic integer is $$ \sum\limits_{n=0}^\infty a_n p^n, \quad a_n\in\{0,1,\ldots, p-1\}. $$ For a $g$-adic integer it is of the form $$ \sum\limits_{n=0}^\infty a_n g^n, \quad a_n\in\{0,1,\ldots, g-1\}. $$ Polyadic integers are of the form $$ \sum\limits_{n=0}^\infty a_n n!, \quad a_n\in\{0,1,\ldots, n\}. $$The main purpose of this work is to estimate from below the correspponding norm of the elements, which is the result of substitution of $p$-adic, $g$-adic or polyadic integers for the variables into a non-zero polynomial with integer coefficients.Therefore, in the case of polyadic integers, we prove the global transcendence and global algebraic independence.Note that when we evaluate the usual absolute value of the considered polynomial, the main difficulty arises to prove the nonvanishing of this polynomial at the approximating point.In $p$-adic, $g$-adic, polyadic cases we avoid it using a well known algebraic lemma on the values of roots of the polynomial.Besides the paper gives some generalization of a theorem by P. Erdös on representation of real number as a sum of two Liouvillean numbers to the cases of $p$-adic, $g$-adic and polyadic numbers.

Détail

Keywords: $p$-adic integer, $g$-adic integer, polyadic integer, estimates of polynomials.

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