Limitations to equidistribution in arithmetic progressions

In the spirit of the landmark work of Granville and Soundararajan, we further develop the theory of limitations to equidistribution in arithmetic progressions for a general class of nonnegative arithmetical functions. Our framework can be applied to numerous specific functions of interest to obtain new results. In particular, we obtain average results on irregularities in equidistribution for sequences of Beatty primes, integers free of small primes, as well as the divisor function supported on integers free of small primes. We also obtain limitations to equidistribution of primes in short arithmetic progressions.