Sequences of algebraic integers and density modulo 1
Let \(\lambda,\mu\) be two algebraic integers such that all their powers \(\lambda^k,\mu^\ell\quad (k,\ell\geq 1)\) are of degree \(d\geq 2\) and distinct. Suppose further that \(\lambda,\mu\) have modulus greater than \(1\), and that \(\mu\) can be written as a polynomial in \(\lambda\) having integer coefficients, and that for the conjugates \(\lambda_i\neq \lambda\) and \(\mu_i\neq \mu\quad(i=2,\dots,d)\) of \(\lambda,\mu\) we have \(|\lambda_i|\neq 1\) and \(\max_i(|\lambda_i|,|\mu_i|)>1\). Then it is proved that for any \(\xi\neq 0\) and sequence of real numbers \(r_m\) the sequence \(\{\mu^m\lambda^n\xi+r_m:n,m\in\mathbb N\}\) is dense modulo \(1\). Earlier Kra had proved the corresponding result in the case \(d=1\), except that \(\xi\) was necessarily restricted to being irrational. The author suspects that the assumption that \(\mu\) is a polynomial in \(\lambda\) with integer coefficients is not necessary. Reviewer's remark: On the other hand, the condition \(\max_i(|\lambda_i|,|\mu_i|)>1\) cannot be removed. Otherwise one could take \(d=2\), \(\lambda=\frac{3+\sqrt{5}}{2}\), \(\mu=2\lambda\), \(\lambda_2=1/\lambda<1\), \(\mu_2=2/\lambda<1\), \(\xi=1\) and \(r_m=0\) for all \(m\). Then since \(\mu^m\lambda^n+ \mu_2^m\lambda_2^n\) is an integer, the only limit point of the sequence \(\{\mu^m\lambda^n\xi+r_m\}\) in this case is the point \(1\).
scientific article; zbMATH DE number 5302622