Simultaneous Approximation of Polynomials

Let P-d denote the family of all polynomials of degree at most d in one variable x, with real coefficients. A sequence of positive numbers x(1) <= x2 <=... is called P-d-controlling if there exist y(1), y(2),....is an element of R such that for every polynomial p is an element of P-d there exists an index i with |p(xi) - yi| <= 1. We settle a problem of Makai and Pach (1983) by showing that x(1) <= x(2) <= ... is P-d- controlling if and only if Sigma(infinity)(i=1) 1/x(i)(d) is divergent. The proof is based on a statement about covering the Euclidean space with translates of slabs, which is related to Tarski's plank problem.