Remarks on certain Salem numbers (Q5947048)

The paper deals with some relations between Salem and Pisot numbers. Let \(\tau\) be a Salem number. The author proves that there are infinitely many positive integers \(n\) such that \(\tau^n-1\) is a Pisot number. From Kronecker's theorem, it is also derived that, for \(\tau\) of degree \(d,\) the smallest such \(n\) belongs to the interval \((d-1) \log 2 / (2 \log \tau) < n < 3^d.\) A Salem number \(\tau\) such that \(\tau-1\) is a unit is called exceptional. The author shows that, if \(\theta\) is a totally positive Pisot unit such that \(\theta-1\) is not a unit, then \(\theta\) is a limit of a sequence of exceptional Salem numbers. This implies that there are arbitrary large exceptional Salem numbers. Furthermore, it is shown that there are arbitrary large Salem numbers \(\tau\) such that \(\tau+1\) is a unit. However, if \(\tau-k\) is a unit, where \(\tau\) is a Salem number of degree \(d\) and where \(k \geq 2\) is a positive integer, then \(|\tau-k|< 1/(2^{d/2}-1).\) Finally, the author asks whether there is a natural number \(k \geq 2\) for which there are infinitely many Salem numbers \(\tau\) such that \(k \tau -k-1\) is a unit.