Algebraic numbers close to 1 in non-Archimedean metrics

Let alpha not equal 1 be an algebraic number with relatively small height. Recently, many authors, including Amoroso, Dubickas, Mignotte and Waldschmidt, stated sharp lower bounds for the quantity \alpha - 1\. Here, we provide a p-adic analogue of their results. For instance, we give an upper bound for the absolute value of the norm of alpha - 1, and we show that our estimate is rather sharp in terms of the degree of alpha. Further, we discuss a generalization in several variables of our result.