On a conjecture on Romanoff type sumsets

Technical Summary of "On a conjecture on Romanoff type sumsets"

Problem Statement
The paper addresses a conjecture by Y.-G. Chen regarding the structure of Romanoff-type sumsets. Let $R$ be the set of odd integers representable as the sum of a prime and a power of two ($p + 2^m$). Romanoff's theorem (1934) established that $R$ has positive lower density. Conversely, van der Corput and Erdős showed that the complement set $N = \mathbb{N}_{odd} \setminus R$ is infinite and contains arithmetic progressions.

The specific problem concerns the set $R_A$, defined as the sumset of primes and a set $A$ generated by powers of two with general exponents:
$$A = \left\{ 2^{\lfloor m_1 r_1 \rfloor} + \dots + 2^{\lfloor m_t r_t \rfloor} : m_i \in \mathbb{N} \right\}$$
$$R_A = P + A = \{ p + a : p \in P, a \in A \}$$
where $r_1, \dots, r_t \ge 1$ are positive numbers satisfying $\sum_{i=1}^t r_i^{-1} \ge 1$.

Chen's Conjecture 1.1 posits that if the exponents are positive integers satisfying the sum condition, the intersection $R_A \cap (R_A + 2)$ (the set of odd numbers $n$ such that both $n$ and $n+2$ are in $R_A$) has a positive lower asymptotic density. While it is known unconditionally that $R_A$ is large, proving that it contains infinitely many pairs of integers with difference 2 (and specifically that such pairs have positive density) remains difficult. The authors note that even proving the set contains infinitely many such pairs unconditionally is currently out of reach.

Methodology
The authors employ a conditional approach, assuming a "Weak uniform Hardy-Littlewood Conjecture" (Conjecture 1.2). This conjecture asserts a uniform lower bound on the number of prime pairs $(p, p+h)$ for even $h \le x^\delta$.

The core technical contribution of the paper is the generalization of a 1950 result by Erdős concerning the $k$-th moments of Romanoff-type representation functions.

1. Moment Generalization: The authors define $f_A(n)$ as the number of representations of an odd integer $n$ as $p + a$ with $a \in A$. They prove Theorem 2.2, establishing an upper bound for the $k$-th moment:
   $$\sum_{n \le x} f_A(n)^k \ll x (\log x)^{k(\sum r_i^{-1} - 1)}$$
   This generalizes Erdős's result (which covered the case $A = \{2^m\}$ and $k$ fixed) to the case where $A$ is a sum of generalized powers of two and the exponents $r_i$ may be real numbers.
2. Analytic Tools: The proof of the moment bound utilizes:
   • Induction on $k$.
   • Sieve methods (specifically Lemma 3.2) to estimate the number of prime tuples satisfying linear constraints.
   • Estimates on the distribution of $2^{\lfloor m r \rfloor}$ modulo $d$, utilizing exponent pairs and properties of the order of 2 modulo $d$ (denoted $e_2(d)$).
   • Lemmas extending results by Erdős, Turán, and Chen to handle non-integer exponents and specific congruence distributions.
3. Cauchy-Schwarz Argument: To prove the main theorem, the authors apply the Cauchy-Schwarz inequality twice to the sum $\sum f_A(n)f_A(n+2)$. This relates the size of the intersection $R_A \cap (R_A + 2)$ to the fourth moment of the representation function and the correlation between representations of $n$ and $n+2$.

Key Results

• Theorem 1.1: Under the assumption of the Weak uniform Hardy-Littlewood Conjecture, the set $R_A \cap (R_A + 2)$ has a positive lower asymptotic density. This holds for positive real numbers $r_i \ge 1$ with $\sum r_i^{-1} \ge 1$, a broader condition than Chen's original conjecture which required integer exponents.
• Theorem 2.2: The generalized upper bound for the $k$-th moment of the representation function $f_A(n)$, which serves as the primary technical engine for the proof.
• Lemma 2.1: A lower bound for the counting function $A(x)$ allowing for real exponents, showing $A(x) \gg (\log x)^{\sum r_i^{-1}}$.

Significance and Claims
The paper claims to provide a "conditional proof" of Chen's conjecture. The authors emphasize that their result covers a wider range of parameters than the original conjecture by allowing the exponents $r_i$ to be real numbers rather than just integers.

The authors are modest regarding the unconditional status of the problem. They explicitly state that they do not know how to prove unconditionally that $R_A \cap (R_A + 2)$ contains infinitely many odd numbers. They frame their work as a step toward resolving the conjecture by demonstrating that, under standard conjectures regarding prime gaps (Hardy-Littlewood), the set is not only infinite but has positive density.

The paper concludes by highlighting the difficulty of the unconditional case, posing Problem 1: to prove unconditionally that the number of such pairs grows faster than $x / \log x$. The authors view their conditional result as a necessary step that isolates the difficulty to the distribution of prime pairs rather than the additive structure of the sumsets.