In 1932 K. Mahler [J. Reine Angew. Math. 166, 118–136 (1931; Zbl 0003.15101, JFM 57.0242.03), ibid. 166, 137–150 (1932; Zbl 0003.38805, JFM 58.0207.01)] introduced a classification of real numbers \(x\) into the so-called classes of \(A, S, T,\) and \(U\) numbers according to the behavior of \(w_n(x)\) defined as the supremum of \(w > 0\) for which \(| P(x)| < H(P)^{-w}\) holds for infinitely many \(P\in P_n\). (Here \(P_n\) denotes the set of integral polynomials of degree \(\leq n\), and \(H(P)\) denotes the height of \(P\), i.e. the maximum of the absolute values of its coefficients.) By Minkowski’s theorem on linear forms, one readily shows that \(w_n(x) > n\) for all \(x\in\mathbb R\). Mahler [Math. Ann. 106, 131–139 (1932; Zbl 0003.24602, JFM 58.0206.04)] proved that for almost all \(x\in\mathbb R\) (in the sense of Lebesgue measure) \(w_n(x)\leq 4n\), thus almost all \(x\in\mathbb R\) are in the \(S\)-class. Mahler has also conjectured that for almost all \(x\in\mathbb R\) one has the equality \(w_n(x)=n\). For about 30 years the progress in Mahler’s problem was limited to \(n=2\) and 3 and to partial results for \(n > 3\). V. Sprindzhuk proved Mahler’s conjecture in full.
Let \(W_n(\Psi)\) be the set of \(x\in\mathbb R\) such that there are infinitely many \(P\in P_n\) satisfying \[ | P(x)| < \Psi(H(P)). \tag{1} \] A. Baker [Proc. R. Soc. Lond., Ser. A 292, 92–104 (1966; Zbl 0146.06302)] has improved Sprindzhuk’s theorem by showing that \(| W_n(\Psi)| = 0\) if \(\sum_{h=1}^\infty \Psi^{1/n}(h) < \infty\) and \(\Psi\) is monotonic.
He also conjectured a stronger statement proved by V. I. Bernik [Acta Arith. 53, 17–28 (1989; Zbl 0692.10042)] that \(| W_n(\Psi)| = 0\) if the sum \[ \sum_{h=1}^\infty h^{n-1}\Psi(h) \tag{2} \] converges and \(\Psi\) is monotonic. Later the author [Acta Arith. 90, No. 2, 97–112 (1999; Zbl 0937.11027)] has shown that \(| \mathbb R\setminus W_n(\Psi)| = 0\) if (2) diverges and \(\Psi\) is monotonic. Here the author proves the following:
Theorem 1. Let \(\Psi: \mathbb R\to \mathbb R^+\) be an arbitrary function (not necessarily monotonic) such that the sum (2) converges. Then \(| W_n(\Psi)| = 0.\)
The author remarks that Theorem 1 is no longer improvable as, by the result of his paper (loc. cit.), the convergence of (2) is crucial.
Theorem 1 of this paper can be readily generalized for non-degenerate curves: Given a non-degenerate map \(f : I \to\mathbb R^n\) defined on an interval \(I\), for any function \(\Psi: \mathbb R\to\mathbb R^+\) such that the sum (2) converges for almost all \(x\in I\) the point \(f(x)\) is not \(\Psi\)-approximable.