[Paper Review] On an inverse ternary Goldbach problem

Xuancheng Shao|arXiv (Cornell University)|Apr 24, 2014

Analytic Number Theory Research10 references3 citations

TL;DR

This paper establishes an inverse ternary Goldbach-type result: for sufficiently large $N$ and small $c>0$, if three subsets $A_1, A_2, A_3 \subset [N]$ each have size at least $N^{1/3 - c}$, then their sumset $A_1 + A_2 + A_3$ must contain a composite number. The proof combines a novel inverse sieve result under additive structure assumptions with a tailored variant of the analytic large sieve, improving upon the prior $N^{1/3 + o(1)}$ bound via Gallagher's larger sieve.

ABSTRACT

We prove an inverse ternary Goldbach-type result. Let $N$ be sufficiently large and $c>0$ be sufficiently small. If $A_1,A_2,A_3\subset [N]$ are subsets with $|A_1|,|A_2|,|A_3|\geq N^{1/3-c}$, then $A_1+A_2+A_3$ contains a composite number. This improves on the bound $N^{1/3}$ from Gallagher's larger sieve. The main ingredients in our argument include a type of inverse sieve result in the larger sieve regime, and a variant of the analytic large sieve inequality.

Motivation & Objective

• To resolve the inverse ternary Goldbach problem by showing that large subsets of $[N]$ cannot sum exclusively to primes.
• To improve the known threshold for subset sizes in ternary sumset problems beyond the $N^{1/3 + o(1)}$ bound from Gallagher's larger sieve.
• To develop a new inverse sieve result in the larger sieve regime, leveraging additive structure in residue classes modulo small primes.
• To construct and apply a variant of the analytic large sieve inequality tailored for sparse sets with structured residue distributions.
• To establish a quantitative bound on the size of sets whose sumsets avoid composite numbers, under structural constraints on their modular distribution.

Proposed method

• Introduce a refined inverse sieve result: if a set $A \subset [N]$ has its reductions modulo each prime $p \leq N^{\alpha}$ contained in an arithmetic progression of length $\alpha p$, then $|A| \ll N^{\alpha - c}$ for small $c > 0$.
• Use a recursive construction of nested sets $A_i$ with decreasing size and progressively smaller residue class coverage, maintaining control over the sum of $\log p / p \cdot |S_p^i|/p$.
• Apply a variant of the analytic large sieve inequality that replaces the standard $\delta$-spacing condition with a weighted sum over primes, enabling tighter bounds when $P \ll N^{1/2}$.
• Leverage additive combinatorics tools, including a standard argument on popular differences, to find a shift $h$ such that $A_i \cap (A_i + h)$ has large intersection and reduces residue class coverage.
• Use a contradiction argument: if no such $h$ reduces the weighted sum of residue class sizes, then the set $H$ of such shifts must be large, leading to a contradiction via the larger sieve.
• Apply Pollard’s theorem and the larger sieve to bound $|H|$, contradicting the lower bound derived from additive energy, thus proving the existence of a good $h$.

Experimental results

Research questions

• RQ1 Can the threshold $N^{1/3 + o(1)}$ for ternary sumsets avoiding composites be improved using structural assumptions on residue classes?
• RQ2 To what extent can the larger sieve be strengthened when the set in question has additive structure, such as being contained in arithmetic progressions modulo small primes?
• RQ3 Is it possible to construct a variant of the analytic large sieve that performs better when the number of points is small relative to $N^{1/2}$?
• RQ4 Does the inverse large sieve conjecture hold for $\alpha > 1/3$, with a nontrivial savings in the size of sets constrained to structured residue classes?
• RQ5 Can the recursive construction of nested sets with decreasing residue class coverage be used to derive stronger uniform bounds in sieve-theoretic settings?

Key findings

• For sufficiently large $N$ and small $c > 0$, any three subsets $A_1, A_2, A_3 \subset [N]$ with $|A_i| \geq N^{1/3 - c}$ must have $A_1 + A_2 + A_3$ contain at least one composite number.
• The bound $N^{1/3 - c}$ improves upon the prior $N^{1/3 + o(1)}$ result obtained via Gallagher’s larger sieve.
• An inverse sieve result is established: if $A \subset [N]$ has its reductions modulo $p \leq N^{\alpha}$ contained in an arithmetic progression of length $\alpha p$, then $|A| \ll N^{\alpha - c}$ for small $c > 0$.
• A new variant of the analytic large sieve inequality is developed, effective when the number of points is much smaller than $N^{1/2}$, enabling tighter bounds in sparse settings.
• The recursive construction of sets $A_i$ with decreasing size and progressively smaller residue class coverage leads to a contradiction if the initial set is too large, proving the main theorem.
• The proof relies on a contradiction derived from the larger sieve applied to the set $H$ of popular differences, showing that such a set cannot be large if the residue class coverage is shrinking fast enough.