  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.
