Approximation properties of polynomials with bounded integer coefficients

Vladimir Drobot and S. McDonald

Vol. 86 (1980), No. 2, 447–450

DOI: 10.2140/pjm.1980.86.447

Abstract

For every fixed positive integea N, let 𝒫_N denote the set of all polynomials p(x) = ∑ a_ixⁱ where a_i is an integer, |a_i|≦ N. For a fixed real number t set 𝒫_N(t) = {p(t) : p ∈𝒫_N}.

Theorem 1. Suppose 1 < t < N + 1 and t is not a root of map of the polynomials from 𝒫_N. Then 𝒫_N(t) is dense in R.

Theorem 2. If t is an S-number then 𝒫_N(t) is discrete for every N.

Mathematical Subject Classification

Primary: 12A15, 12A15

Milestones

Received: 10 November 1978

Revised: 4 May 1979

Published: 1 February 1980