An Approximation Property of Pisot Numbers Original Research Article

Let q>1. Initiated by P. Erdimages et al. in [4], several authors studied the numbers lm(q)=inf{y: yset membership, variantΛm, y≠0}, m=1, 2, …, where Λm denotes the set of all finite sums of the form y=var epsilon0+var epsilon1q+var epsilon2q2+…+var epsilonnqn with integer coefficients −mless-than-or-equals, slantvar epsiloniless-than-or-equals, slantm. It is known ([1], [4], [6]) that q is a Pisot number if and only if lm(q)>0 for all m. The value of l1(q) was determined for many particular Pisot numbers, but the general case remains widely open. In this paper we determine the value of lm(q) in other cases.