Comments on the height reducing property

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F , of the ring ℤ of the rational integers such that ℤ[ α ] = F [ α ]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.