Integer Addition and Hamming Weight

We study the effect of addition on the Hamming weight of a positive integer. Consider the first 2n positive integers, and fix an α among them. We show that if the binary representation of α consists of Θ(n) blocks of zeros and ones, then addition by α causes a constant fraction of low Hamming weight integers to become high Hamming weight integers. This result has applications in complexity theory to the hardness of computing powering maps using bounded-depth arithmetic circuits over F2. Our result implies that powering by α composed of many blocks require exponential-size, bounded-depth arithmetic circuits over F2.