Simultaneous Approximation of Polynomials
نویسندگان
چکیده
Let Pd denote the family of all polynomials of degree at most d in one variable x, with real coefficients. A sequence of positive numbers x1 ≤ x2 ≤ . . . is called Pd-controlling if there exist y1, y2, . . . ∈ R such that for every polynomial p ∈ Pd there exists an index i with |p(xi)−yi| ≤ 1. We settle an problem of Makai and Pach (1983) by showing that x1 ≤ x2 ≤ . . . is Pd-controlling if and only if ∑︀∞ i=1 1 xi 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.
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