Some subcritical estimates for the $\ell^p$-improving problem for discrete curves

Abstract

We apply Christ's method of refinements to the ℓ p -improving problem for discrete averages A N along polynomial curves in Z d . Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations, we obtain some restricted weak-type p → p ′ estimates for the averages A N in the subcritical regime. The dependence on N of the constants here obtained is sharp, except maybe for an ǫ-loss.

Subcritical paucity and $\ell^p$-improving estimates for finite-type polynomial curves

“…Moreover, (1.1) is true at the exponents pp, qq " p1, 8q and pp, qq " p2, 2q as shown by Young's inequality and Plancherel's theorem respectively. Note that my definition of subcritical vs supercritical differs from [DHV20+] to be more in line with the use for a subcritical vs supercritical number of variables in the underlying system of Diophantine equations that I will study.…”

“…The arithmetic method of refinements in [DHV20+] (and Section 2) permits us to establish a relationship between the study of inhomogeneous systems of Diophantine equations and subcritical ℓ p -improving estimates. Fix γ " pφ 1 , .…”

“…I do not attempt to survey this large body of works. Instead, I will mention that, to the best of my knowledge, this method has been used in the study of discrete fractional integrals and discrete ℓ p -improving estimates in [Obe01,Pie12,Kim15,DHV20+].…”

“…A crucial difference in [DHV20+] compared to earlier works is the novel use of the method of refinements to prune special subvarieties from the analysis and reduce the problem to proving paucity estimates for inhomogeneous systems of equations. In turn, [DHV20+] used Wooley's second order differencing method from [Woo91,Woo93,HW21] to prove new paucity estimates for inhomogeneous systems of equations.…”

“…My approach follows the one introduced in [DHV20+]. In particular, Conjecture 1.2 is a direct generalization of Conjecture 1 in [DHV20+] and Theorem 1.6 is a generalization of Theorem 1 in [DHV20+]. I adopt a more modular approach that explicitly states Theorem 1.3.…”