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- W2964293876 abstract "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 $(t,talpha,...,talpha^{m-1})$ for $tinR$ 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." @default.
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- W2964293876 date "2012-01-01" @default.
- W2964293876 modified "2023-10-18" @default.
- W2964293876 title "An equivalent of Kronecker's theorem for powers of an algebraic number and structure of linear recurrences of fixed length" @default.
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- W2964293876 doi "https://doi.org/10.4064/aa153-1-2" @default.
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