Sparse Bounds for Discrete Maximal Functions associated with Birch-Magyar averages

In this article, we study discrete maximal function associated with the Birch-Magyar averages over sparse sequences. We establish sparse domination principle for such operators. As a consequence, we obtain ${\mathrm{\ell }}^p$-estimates for such discrete maximal function over sparse sequences for all $p>1$. The proof of sparse bounds is based on scale-free ${\mathrm{\ell }}^p-$improving estimates for the single scale Birch-Magyar averages.