Approximation of real matrices by integral matrices (Q5899780)

The author proves the following result. Let \((\alpha_{ij})\) be a real \(k\times k\) matrix for \(k\geq 2\) with \(\det \alpha_{ij}=1\). Then there is a constant \(C=C(\alpha_{ij})\) such that for every positive integral \(k\times k\) matrix \((A_{ij})\) with det \(A_{ij}=n\) and \[ \max_{1\leq i,j\leq k}| A_{ij}-n^{1/k}\alpha_{ij}| <Cn^{3/4k}. \] The exponent 3/4k cannot be replaced by any number \(<1/2k.\) This result improves upon and extends a previous result due to \textit{R. Tijdeman} [J. Number Theory 24, 65-69 (1986; Zbl 0595.10026)] who considered the case \(k=2\). The author's method is different from the previous work on this problem. He applies \textit{A. Weil}'s estimate for the Kloosterman sum [Proc. Natl. Acad. Sci. USA 34, 204-207 (1948; Zbl 0032.26102)] and a result on primes in short intervals by \textit{M. N. Huxley} [Invent. Math. 15, 164-170 (1972; Zbl 0241.10026)].