Small remainder of a vector to suitable modulus
Let \(R\) be a natural number and \({\mathbf x}\) an integer vector in \(t\) dimensions with \(|{\mathbf x}|\leq R^K\). There are positive constants \(c= c(K)\) and \(C_j= C_j (K, t)\) with the following property. Let \(C_1\leq Q\leq R^c\). Let \(\psi_1, \dots, \psi_t\) be positive numbers with \(\psi_i\leq C_2^{-1} (\log Q)^{-t}\) \((i=1,\dots,t)\), \(\psi_1 \dots \psi_t= Q^{-1}\). Then \(|{x_i} \pmod r|\leq r\psi_i\) (\(i=1,\dots, t)\) for some integer \(r\), \(R\leq r\leq 2RQ\). Examples are given to show that the range of \(r\) is essentially as short as possible. (Bombieri has an application to an effective sharpening of Liouville's theorem in diophantine approximation; we do not give details of this application, which are not available yet.) The key to the proof of our theorem is a `lattice method' of W. M. Schmidt.
scientific article; zbMATH DE number 861934
scientific article