arXiv subjects
Publications and source records attributed to Toufik Zaimi.
We show that for any natural number $n$ satisfying $n\equiv 4 \mod 8$ and $n\not\equiv 0 \mod 5$, and for any odd integer $t\geq \frac{n+6}{2}$ there are infinitely many Salem numbers $α$ of degree $2t$ such that $α^n-1$ is a unit. This result, obtained using a generalization of a construction due to Gross and McMullen [5], partially completes the main result of [7].
A well-known result, due to Meyer, states that the set P of Pisot numbers, generating a real algebraic number field K, is uniformly discrete and relatively dense in the set of positive real number. In the present paper, we show that P is contained is the set, say D, of the differences of elements of P, and the complement of P in D is not finite. Also, we prove that if K is totally real, then the elements of D are the algebraic integers of K whose images under the action of all embeddings of K into R, other than the identity of K, belong to the interval (-2,2).
By extending a construction due to Gross and McMullen [2], we show that for any odd integer n and for any even integer d>n+2 there are infinitely many Salem numbers $α$ of degree d such that $α^n-1$ is a unit. A similar result is also proved when n runs through some classes of even integers, d>n+3 and d/2 is an odd integer.
A complex number alpha is said to satisfy the height reducing property if there is a finite subset F of the ring Z of the rational integers such that Z[alpha]=F[alpha]. This problem of finding F has been considered by several authors, especially in contexts related to self affine tilings, and expansions of real numbers in non-integer bases. We continue, in this note, the description of the numbers satisfying the height reducing property, and we specify a related characterization of the roots of integer polynomials with dominant term.