On the distribution of the van der Corput sequences

Abstract

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