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

Hickman and Wright proved an L² restriction estimate for the parabola $Σ$ in Z/NZ of the form (1/|Σ|∑m∈Σ|f(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|f(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|f(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.