Uniform approximation by polynomials with integer coefficients

Artur Lipnicki

Abstract. Let \(r\), \(n\) be positive integers with \(n\ge 6r\). Let \(P\) be a polynomial of degree at most \(n\) on \([0,1]\) with real coefficients, such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\). It is proved that there is a polynomial \(Q\) of degree at most \(n\) with integer coefficients such that \(|P(x)-Q(x)|\le C_1C_2^r r^{2r+1/2}n^{-2r}\) for \(x\in[0,1]\), where \(C_1\), \(C_2\) are some numerical constants. The result is the best possible up to the constants.

Keywords: approximation by polynomials with integer coefficients, lattice, covering radius.

Mathematics Subject Classification: 41A10, 52C07.

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