Restriction estimates with sifted integers

Technical Summary: Restriction Estimates with Sifted Integers

Problem Statement
The paper addresses the problem of obtaining restriction estimates for exponential sums over sifted sequences of integers. While Bourgain established upper bounds for the $L^\ell$-norm of exponential sums over primes, and Green-Tao generalized these to broader arithmetic sets using enveloping sieves, this work specifically targets sequences $S(N)$ defined by sieving conditions. The set $S(N)$ consists of integers $n \le N$ such that $n \pmod p \notin L_p$ for a given subset of primes $P$ and subsets $L_p \subset \mathbb{Z}/p\mathbb{Z}$. The goal is to derive estimates of the form:
$$\int_0^1 \left| \sum_{n \in S(N)} a_n e(n\alpha) \right|^\ell d\alpha \le C \left( \frac{N}{V(z)} \sum_{n \in S(N)} |a_n|^2 \right)^{\ell/2}$$
for $\ell > 2$, where $V(z)$ represents the density of the sifted set. A critical aspect of the problem is the behavior at the endpoint $\ell=2$, where standard restriction estimates fail, and the authors aim to provide a smooth transition from $\ell > 2$ to $\ell = 2$.

Methodology
The authors employ a combination of the enveloping sieve method and duality arguments, following the framework established by Ramaré and Ruzsa.

1. Enveloping Sieve Construction: Instead of analyzing the characteristic function of the sifted set $S(N)$ directly, the authors construct a majorant function $\beta$ with favorable Fourier properties. This majorant is derived using the enveloping sieve of Ramaré and Ruzsa, which allows the characteristic function of the sifted set to be bounded by a sum of exponential terms with controlled coefficients.
2. Dual Large Sieve Inequality: A central component of the proof is a dual large sieve type inequality (Theorem 1.3). This inequality bounds the sum of squares of exponential sums over a $\delta$-well-spaced set $B \subset \mathbb{R}/\mathbb{Z}$, specifically:
   $$\sum_{n \in S(N)} \left| \sum_{b \in B} f(b) e(nb) \right|^2 \le K \frac{N + \delta^{-1}}{V(z)} \log^\kappa \left( \frac{2\|f\|_1^2}{\|f\|_2^2} \right) \|f\|_2^2$$
   This result relies on the specific structure of the sieve weights and the assumption that the cardinality of the forbidden sets $L_p$ satisfies $\lambda(p) \le M p^{1/4}$.
3. Interpolation and Summation: To move from the $L^2$ estimate (Theorem 1.4) to the $L^\ell$ restriction estimate (Theorem 1.2), the authors utilize a technique involving the distribution of large values of the exponential sum. They define level sets $B_\xi$ and apply the Cauchy-Schwarz inequality combined with the dual large sieve bound. The summation over these level sets is managed using properties of Eulerian polynomials (Carlitz identity) to handle the dependence on $\ell$.

Key Contributions and Results
The paper presents three main theorems:

• Theorem 1.3 (Dual Large Sieve): Establishes a large sieve inequality for sifted integers with an explicit dependence on the ratio of $L^1$ and $L^2$ norms of the test function. This is described as the "heart" of the paper.
• Theorem 1.4 (Large Sieve for Sifted Integers): Derives a standard large sieve inequality for the sequence $S(N)$ using Theorem 1.3 and Cauchy-Schwarz.
• Theorem 1.1 (Restriction Estimate): Provides the main restriction estimate for the $L^\ell$-norm of exponential sums over $S(N)$ for $\ell > 2$. The constant $C$ is explicitly computable and depends on the sieve parameters $\kappa, T, M$ and the exponent $\ell$.
• Hardy-Littlewood Majorant Property (Theorem 7.1): Under the assumption that the size of the sifted set $|S(N)|$ is comparable to $N/V(z)$, the authors show that the sequence $S(N)$ enjoys the Hardy-Littlewood majorant property. This implies that the $L^\ell$ norm of the exponential sum with arbitrary coefficients is bounded by the norm of the sum with coefficients of modulus 1.

Significance and Claims
The authors claim that their results generalize the restriction estimates of Green-Tao, which were originally formulated for primes and specific arithmetic sets, to a wider class of sifted integers defined by arbitrary subsets $L_p$.

A significant technical feature highlighted by the authors is the "smooth transition" from $\ell > 2$ to $\ell = 2$. While the restriction estimate fails at $\ell=2$ due to Parseval's identity, the explicit dependency of the constant on $\ell$ (derived in equations 6.2 and 6.3) allows for a controlled approach to the endpoint, a feature attributed to the method of Ramaré.

The paper demonstrates the utility of these estimates through three applications:

1. Recovering the restriction estimates for polynomial values (Green-Tao's Theorem 1.1) as a corollary of their general framework.
2. Improving the upper bound for exponential sums related to integers composed of primes congruent to $3 \pmod 4$ (Application 2), specifically removing a $\sqrt{\log z}$ factor present in previous bounds by Harper.
3. Providing estimates for the set of integers that are sums of two coprime squares and satisfy a shift condition (Application 3).

The work relies heavily on the enveloping sieve developed by Ramaré and Ruzsa, extending their machinery to derive explicit constants and broader applicability for restriction problems in analytic number theory.