Mathematics

For an integer $p\ge 2$, let $\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$ be the p-adic van der Corput sequence. For intervals $[0,\alpha )\subset {\mathbb {T}}$ and for positive integers N, consider the geometrically-shifted discrepancy function $D_{p,N,\alpha }(t)=\sum _{n=0}^{N-1}\mathcal {X}_{[0,\alpha )}(x_n+t)-N\alpha$. In this paper, we give a characterization of the asymptotic behavior of $\Vert D_{p,N,\alpha }(\cdot )\Vert _{L^2({\mathbb {T}})}$ for $N\rightarrow \infty$that depends on the p-adic expansion of $\alpha$.