On Comparing Sums of Square Roots of Small Integers

Abstract

Let k and n be positive integers, n>k. Define r(n,k) to be the minimum positive value of \( |\sqrt{a_1} + \cdots + \sqrt{a_k} -- \sqrt{b_1} -- \cdots -\sqrt{b_k} | \) where a 1, a 2, ⋯, a k , b 1, b 2, ⋯, b k are positive integers no larger than n. It is an important problem in computational geometry to determine a good upper bound of –logr(n,k). In this paper we prove an upper bound of 2\(^{O({\it n}/log{\it n})}\), which is better than the best known result O(22k logn) whenever n ≤ck logk for some constant c. In particular, our result implies an algorithm subexponential in k (i.e. with time complexity 2\(^{o({\it k})}\)(logn)O(1) ) to compare two sums of square roots of integers of value o(k logk).