waring-intervals

The intervals for Waring’s problem with $n = 2^k,~ 3^k,~ 4^k$ respectively on unit circles in the complex plane are shown by the sections in pink, which are the points $\e^{2\pi \ii\alpha}$ satisfying $\left| \alpha – a/q \right| < n^{-1 + 1/(5k)}$ for each $1\leq q\leq n^{1/k}$ and $0\leq a/q\leq 1$. The intervals with smaller contributions are the leftover sections in blue. Notice how the intervals shrink as there are more and more disjoint intervals. In reality the pink sections are much much smaller than what is shown here but then they would be too difficult to see!