Three Examples of Triangular Arrays with Optimal Discrepancy and Linear Recurrences

Abstract

In the first part of this short note we give an answer to a question asked by Franz J. Schnitzer (Leoben). Let ${({\alpha _k})_{k \geqslant 1}}$ denote the sequence of positive zeroes of the Bessel function J ₀(x) in increasing order. We consider the triangular arrays $({x_{kN}}) = \left( {\frac{{{\alpha _k}}}{{{\alpha _N}}}} \right),1 \leqslant k \leqslant N,N \in N$ in [0,1). F.J. Schnitzer (personal communication) has conjectured that this triangular array is uniformly distributed modulo 1, i.e.

$$ \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\# \{ k \leqslant N:{x_{kN}} \in I\} = \left| I \right| $$

((1))

for any subinterval $I \subseteq [0,1)$ of length ∣I∣.