Sums of monomials with large Mahler measure

For n ? 1 let A n ? { P : P ( z ) = ? j = 1 n z k j : 0 ? k 1 < k 2 < ? < k n , k j ? Z } , that is, A n is the collection of all sums of n distinct monomials. These polynomials are also called Newman polynomials. If α < β are real numbers then the Mahler measure M 0 ( Q , α , β ] ) is defined for bounded measurable functions Q ( e i t ) on α , β ] as M 0 ( Q , α , β ] ) ? exp ( 1 β - α ? α β log | Q ( e i t ) | d t ) . Let I ? α , β ] . In this paper we examine the quantities L n 0 ( I ) ? sup P ? A n M 0 ( P , I ) n and L 0 ( I ) ? lim inf n ? ∞ L n 0 ( I ) with 0 < | I | ? β - α ? 2 π .