Binary Sequences Constructed by the Truncated Sum-of-Digits Function of Squares

We introduce a family of pseudorandom binary sequences defined by the truncated sum-of-digits function of squares in base-q with q≥2. Using the Erdős–Turán inequality and Weyl-type bounds for quadratic exponential sums, we estimate the well-distribution measure and the correlation measure of order l for these sequences. For k=2 (i.e., using the first two base-q digits), we prove that W(Eq2)≪q(logq)2 and Cl(Eq2)≪2lq(logq)l+1, provided that q is sufficiently large. These bounds show that the sequences exhibit strong pseudorandom properties.