Rational approximation to orthogonal bases and of the solutions of elliptic equations

It is shown that given N greater than or equal to 2, an orthogonal basis phi(1), ..., phi(n) of R-n can be approximated by an orthogonal basis b(1),..., b(n), where b(1) has integral and b(2),..., b(n) have rational components, such that the angle between phi(i) and b(i) is at most 1/N and the length \b(i)\ less than or equal to 10n(3)N, i = 1, ..., n. This improves the length of the integral approximation due to Schmidt (1995). As an application, we improve a theorem of Kocan (1995) about the minimal size of grids in the solutions of elliptic equations. Our result fits the need in Kuo and Trudinger(1990). Mathematics Subject Classification (1991): 11H06, 35A40.