On Salem numbers which are exceptional units

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 $\alpha$ of degree d such that ${\alpha }^n-1$ 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.