Product type bounds on the approximation of values ofE andG functions

This paper obtains effective lower bounds on the absolute values of linear forms, over the integers, in power products of values of certain SiegelE-functions or SiegelG-functions. The bounds obtained are in terms of the product of the absolute values of the coefficients. ForE-functions the bound obtained is a best possible result, up to an arbitrarily small positive epsilon. ForG-functions the result is asymptotically best in the following sense: for each epsilon larger than zero there exists an integerN such that ifz, the point of evaluation, equalsM ^–1 whereM is an integer with absolute value larger thanN, then the bound obtained is within epsilon of a best possible bound. (From the proof it is clear thatz need not be the inverse of an integer. What is necessary that the absolute value of its numerator must be much smaller than the absolute value of its denominator.)Results obtained recently by D. V. andG. V. Chudnovsky bounding the absolute values of similar forms give bounds in terms of the maximum of the absolute values of the coefficients; such lower bounds can be much smaller.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday