Discrepancy Results for The Van Der Corput Sequence

Let dN = NDN (ω) be the discrepancy of the van der Corput sequence in base 2. We improve on the known bounds for the number of indices N such that dN ≤ log N/100. Moreover, we show that the summatory function of dN satisfies an exact formula involving a 1-periodic, continuous function. Finally, we give a new proof of the fact that dN is invariant under digit reversal in base 2.

Onnth-order differential operators with Bohr-Neugebauer type property

SupposeBis a bounded linear operator in a Banach space. If the differential operatordndtn−Bhas a Bohr-Neugebauer type property for Bochner almost periodic functions, then, for any Stepanov almost periodic continuous functiong(t)and any Stepanov-bounded solution of the differential equationdndtnu(t)−Bu(t)=g(t),u(n−1),…,u′,uare all almost periodic.

On Second-Order Differential Operators With Bohr-Neugebauer Type Property

Let B be a bounded linear operator having domain and range in a Banach space. If the second-order differential operator d2/dt2–B has a Bohr-Neugebauer type property for Bochner almost periodic functions, then any Stepanov-bounded solution of the differential equation (d2/dt2)u(t) – Bu(t) = g(t) is Bochner almost periodic, with g(t) being a Stepanov-almost periodic continuous function.