A two-representation set with logarithmic exponent 3/7

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_proof: A uniform divisor count for fiber triples and simultaneous shell selection give one set with lower exponent 3/7 and an encoding-specific zero ratio.

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For a set $A\subseteq\mathbb Z_{>0}$, define

$$A(x)=|A\cap[1,x]|,\qquad r_A(n)=|\{(a,b)\in A^2:a\le b,\ a+b=n\}|.$$

Thus representations are unordered and a diagonal $a+a$ counts once.

Statement

There is one infinite set $A\subseteq\mathbb Z_{>0}$ such that

$$r_A(n)\le2\quad(n\in\mathbb Z),\qquad \liminf_{x\to\infty}\frac{\log A(x)}{\log x}\ge\frac37.$$

The limit inferior is over real cutoffs. In particular, for every real $\alpha<3/7$ this same set satisfies $A(x)\ge x^\alpha$ at every sufficiently large cutoff. The set is chosen before $\alpha$; only the threshold depends on $\alpha$.

The proof constructs shifted discrete-logarithm digits, bounds triples in the associated congruence fibers uniformly over pairs of shells, and chooses an independent subset of each upper shell using every lower shell at once. It proves the bound between shell cutoffs as well as at them.

Restricted-encoding zero endpoint ratio

Let $A_0$ be the full ambient label set defined in the proof, before any selection, with auxiliary primes $q_j$, radix products $B_K$, and $\gamma=3/7$. At the cutoffs

$$x_K=(q_{K+1}+1)B_K-1$$

one has

$$0\le\frac{A_0(x_K)}{x_K^{3/7}} \le2^{-(3/7)(4K+1)}\longrightarrow0.$$

Consequently every subset $S\subseteq A_0$, including the selected $A$, satisfies $\liminf_{x\to\infty}S(x)/x^{3/7}=0$. This is a consequence about this particular encoding. It does not preclude a different construction with an eventual positive constant times $x^{3/7}$.

Scope and sources

The result bears on E0158, which asks about lower density at the square-root scale. The logarithmic exponent $3/7<1/2$ does not answer that question. No optimality or priority claim is made here.

The method adapts the mixed-radix construction in Javier Cilleruelo's Infinite Sidon sequences, arXiv:1209.0326v2, 15 May 2013, §§2.1–2.2 and Propositions 1–2, pp. 4–6. The auxiliary-prime intervals and label shells used here differ from that version. All digital properties needed for this construction are proved locally. Cilleruelo's theorem is a method relationship, not a premise of L2.

The external theorem inputs are Bertrand's postulate, cyclicity of the multiplicative group of a prime field, and the prime number theorem, at the interfaces stated in the proof. The proof supplies the uniform divisor bound, the congruence and prime-factor arguments, and the finite alteration lemma. It consumes no native L-claim.

Current standing

The complete proof and the restricted-encoding consequence are author-recorded at tier 0. They have not received independent whole-claim acceptance or native Lean verification. There is no identified gap in the supplied argument, and there is no mathematical program or computational input to run.

The author check covered the displayed construction, both determinant cases, the estimates uniform over lower shells, the assembly of the single infinite set, and the restricted-encoding consequence. Independent acceptance would require scrutiny of this exact statement, proof, external interfaces, and named consequence. This record does not claim independently accepted literature-proof coverage.