An equivalent of Kronecker's Theorem for powers of an Algebraic Number and Structure of Linear Recurrences of fixed length
| Autor: | Dubbini, Nevio, Monge, Maurizio |
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| Rok vydání: | 2009 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | After defining a notion of $\epsilon$-density, we provide for any real algebraic number $\alpha$ an estimate of the smallest $\epsilon$ such that for each $m>1$ the set of vectors of the form $(t,t\alpha,...,t\alpha^{m-1})$ for $t\in\R$ is $\epsilon$-dense modulo 1, in terms of the multiplicative Mahler measure $M(A(x))$ of the minimal integral polynomial $A(x)$ of $\alpha$, and independently of $m$. In particular, we show that if $\alpha$ has degree $d$ it is possible to take $\epsilon = 2^{[d/2]}/M(A(x))$. On the other hand using asymptotic estimates for Toeplitz determinants we show that for sufficiently large $m$ we cannot have $\epsilon$-density if $\epsilon$ is a fixed number strictly smaller than $1/M(A(x))$. As a byproduct of the proof we obtain a result of independent interest about the structure of the $\Z$-module of integral linear recurrences of fixed length determined by a non-monic polynomial. Comment: 15 pages, 1 figure |
| Databáze: | arXiv |
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