Tomaszewski'S Problem On Randomly Signed Sums, Revisited

Let v(1), v(2), . . . , v(n) be real numbers whose squares add up to 1. Consider the 2(n) signed sums of the form S = Sigma +/- v(i). Boppana and Holzman (2017) proved that at least 13/32 of these sums satisfy vertical bar S vertical bar <= 1. Here we improve their bound to 0.427685.