$ell^p(mathbb{Z}^d)$-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Means, Revisited

We prove an expanded range of $ell ^{p}(mathbb{Z}^d)$-improving properties and sparse bounds for discrete spherical maximal means in every dimension $dgeq 6$. Essential elements of the proofs are bounds for high exponent averages of Ramanujan and restricted Kloosterman sums.