On the minimum gap between sums of square roots of small integers

Let k and n be positive integers, n > k. Define r(n, k) to be the minimum positive value of | √ a1 + · · ·+ √ ak − √ b1 − · · · − √ bk| where a1, a2, · · · , ak, b1, b2, · · · , bk are positive integers no larger than n. Define R(n, k) to be − log r(n, k). It is important to find tight bounds for r(n, k) and R(n, k), in connection to the sum-of-square-roots problem, a famous open problem in computational geometry. The current best lower bound and upper bound are far apart. In this paper, we prove an upper bound of 2 logn) for R(n, k), which is better than the best known result O(2 log n) whenever n ≤ ck log k for some constant c. In particular, our result implies an algorithm subexponential in k (i.e. with time complexity 2(log n) ) to compare two sums of k square roots of integers of value o(k log k). We then present an algorithm to find r(n, k) exactly in n time and in ndk/2e+o(k) space. As an example, we are able to compute r(100, 7) exactly in a few hours on a single PC. The numerical data indicate that the root separation bound is very far away from the true value of r(n, k).