On the Glasner property of linear maps with prime entries on tori

We study the quantitative Glasner property in the context of maps between tori of differing dimension, instead of as a (semi-)group action. We also only consider matrices with entries being non-constant polynomials evaluated at primes, extending the work of Nair and Velani [Glasner sets and polynomials in primes, Proc. Amer. Math. Soc. 126(10) (1998) 2835–2840] and Bulinski and Fish [Glasner property for unipotently generated group actions on tori, Israel J. Math. 255 (2022) 109–122, doi:10.1007/s11856-022-2412-5] to a more general setting.