On the digits of squares and the distribution of quadratic subsequences of digital sequences

Abstract:

Let $q$ be a prime and $\gamma =(\gamma _0, \gamma _1, \gamma _2, \dots )$ with $\gamma _i\in \{0, \dots , q-1\}$ and $\gamma _i=0$ for $i\geq i_0$ be a weight sequence.

We study $\lim _{N\rightarrow \infty }{\#\left \{0\leq n< N|s_{q,\gamma }(n^2)\equiv d (\text {mod }q)\right \}}/N$ with $s_{q,\gamma }(n^2)$ the weighted sum of digits of $n^2$ in base $q$ and we use the results to classify the digital sequences $\left (\boldsymbol {x}_n\right )_{n\geq 0}$ in the sense of Niederreiter, generated by matrices with finite rows, for which $\left (\boldsymbol {x}_{n^2}\right )_{n\geq 0}$ is uniformly distributed. Finally we derive an upper bound for the star discrepancy of these uniformly distributed subsequences.