On a Restriction Problem of Hickman and Wright for the Parabola in ℤ/Nℤ for Squarefree N

2025/09/11 by Kingsbury-Neuschotz, Nathaniel
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.09885

Abstract

Hickman and Wright proved an L² restriction estimate for the parabola Σ in ℤ/Nℤ of the form ((1)/(|Σ|)∑_m∈Σ|\widehatf(m)|² )^(1)/(2)≤ C_εN^ε⋅ N^-1(∑_{x∈ (ℤ/Nℤ)²}|f(x)|^(6)/(5))^(5)/(6) for all functions f:(ℤ/Nℤ)²→ ℂ and any ε>0, and that this bound is sharp when N has a large square factor, and especially for N = p² for p a prime. In contrast, Mockenhaupt and Tao proved in the special case N = p the stronger estimate ((1)/(|Σ|)∑_m∈Σ|\widehatf(m)|² )^(1)/(2)≤ C N^-1(∑_{x∈ (ℤ/Nℤ)²}|f(x)|^(4)/(3))^(3)/(4). We extend the Mockenhaupt-Tao bound to the case of squarefree N, proving ((1)/(|Σ|)∑_m∈Σ|\widehatf(m)|² )^(1)/(2)≤ C_εN^ε⋅ N^-1(∑_{x∈ (ℤ/Nℤ)²}|f(x)|^(4)/(3))^(3)/(4), and discuss applications of this result to uncertainty principles and signal recovery.

Citations

• Fourier minimization and imputation of time series
• Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings
• A bilinear approach to the finite field restriction problem
• The Square-Root Law Does Not Hold in the Presence of Zero Divisors
• Finite field restriction estimates for the paraboloid in high even\n dimensions
• The square root law and structure of finite rings
• Endpoint restriction estimates for the paraboloid over finite fields
• Extension theorems for the Fourier transform associated with non-degenerate quadratic surfaces in vector spaces over finite fields

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• Sharp superlevel set estimates for small cap decouplings of the parabola
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• New Restriction Estimates for the 3-d Paraboloid over Finite Fields