Erdős Problem #39: independent continuation
Date: 23 September 2026. Source versions read in full:
39.tex in
gpt_pro_5.4.
1) OBJECTIVE AND SOURCE AUDIT
The target is one infinite Sidon set with $A(x)\gg_\varepsilon x^{1/2-\varepsilon}$ for every $\varepsilon>0$. The source treats dense infinite Sidon sequences through a mixed-radix construction. Removing illegal discrete logarithms repairs a definition, but does not itself re-prove the unavailable earlier-round collision estimates. Also an upper exponent $\gamma(c)>c$ shows only that this estimate is insufficient, not that the construction actually fails.
2) FINITE PREFIXES DO NOT OBSTRUCT A GROWTH EXPONENT
Here is an independent extension statement. Let $F\subseteq[1,K]$ be any finite Sidon set, and let $A\subseteq\mathbb N$ be any infinite Sidon set. Put \[
C=2K+1,\qquad M=6K+3,\qquad B=F\cup\{Ma+C:a\in A\}.
\] Then $B$ is Sidon. Modulo $M$, sums from two elements of $F$ lie in $[2,2K]$; mixed sums lie in $[2K+2,3K+1]$; sums from two elements of the new tail have residue $4K+2$. These residue ranges are disjoint and all lie below $M$, so collisions cannot mix the three types. Within $F$ use its Sidon property. Within the tail, divide an equality by $M$ and use the Sidon property of $A$. Two mixed sums satisfy \[
f_1-f_2=M(a_2-a_1),
\] and $|f_1-f_2|<M$ forces both differences to vanish. This also handles repeated summands in the usual Sidon convention. If $A(x)=x^{c+o(1)}$, then \[
B(x)=|F|+A((x-C)/M)=x^{c+o(1)}.
\] Thus every available exponent can be realized while preserving any prescribed finite Sidon prefix.
3) CONSEQUENCE FOR THE SUPPLIED APPROACH
The extension theorem separates finite repairs from the real asymptotic difficulty: finitely many exceptional indices, or a prescribed initial segment, cannot prevent an exponent already achieved by some infinite Sidon sequence. What remains difficult is controlling infinitely many potential collisions while increasing that exponent. This lemma does not supply such control and does not approach exponent $1/2$ by itself.
7) FINAL
UNRESOLVED. A fully proved finite-prefix extension principle is obtained, without claiming a new best density or independent validation of the source's endpoint estimates. Problem record:
https://www.erdosproblems.com/39.
8) COMPLETION ESTIMATE
COMPLETION: 10% This is a subjective estimate of progress toward the entire stated problem, not a probability that the conjecture or an unverified proof is correct.