New bounds for discrete lacunary spherical averages

Abstract

We show that the discrete lacunary spherical maximal function is bounded on l p (Z d ) for all p > d+1 d−1 . Our range is new in dimension 4, where it appears that little was previously known for general lacunary radii. Our technique, using the Kloosterman refinement, also allows us to recover the range of l p improving estimates in all dimensions d ≥ 4. Though our range does not improve on the current known l p bounds for dimensions six and higher, our proof allows us to get estimates for the main term of the multiplier decomposition that match the conjectured overall sharp bounds.

Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

“…4.□ For c ∈ (1, 2) and g ∈ C ∞ c (R) we introduce Π g t,s (ξ) := n∈Z e(⌊|n| c ⌋t + nξ)g n s 1/c , s ≥ 1, t ∈ R, ξ ∈ T,(4.10)and the quantityN c := (2c) −1 (2N ) κ . Let c ∈ (1, 2) and g ∈ C ∞ c (R) be fixed.Then for all N ≥ 1 one has sup (ξ)| ≲ N 1/3+1/(3c) log(N + 1).…”