gpt 5.2 attempt on Math-650

OUTPUT 1) FORMAL RESTATEMENT A set $A\subseteq\mathbb N$ is an additive complement to the primes if every sufficiently large integer $n$ can be written as \[n=p+a\quad\text{with $p$ prime and $a\in A$.}\] Questions (as stated): (a) Can one choose such an $A$ with $|A\cap[1,N]|=o((\log N)^2)$? (b) Can one achieve $|A\cap[1,N]|=O(\log N)$? (c) Must any such $A$ satisfy $\liminf_{N\to\infty} |A\cap[1,N]|/\log N >1$?

2) QUICK LITERATURE/CONTEXT CHECK Browsing is not available. I only record what the problem statement itself claims. I have not verified those results here.

3) ATTACK PLAN Proof track: 1. Use probabilistic method: choose $A$ randomly with density about $(\log N)^2/N$ on dyadic intervals and show coverage. 2. Prove lower bounds on $|A\cap[1,N]|$ by a counting argument using $\pi(N)$.

Disproof track: 1. Search for congruence obstructions preventing $O(\log N)$ complements.

Chosen path: prove an unconditional counting lower bound in terms of $\pi(N)$.

4) WORK Lemma 1 (Counting lower bound via prime counting). Assume every integer $n$ with $1\le n\le N$ can be written as $n=p+a$ with $p$ prime and $a\in A$. Let $m(N):=|A\cap[1,N]|$ and let $\pi(N)$ be the number of primes $\le N$. Then \[m(N)\,\pi(N)\ge N. \]

Proof. For each $a\in A\cap[1,N]$, the integers representable as $p+a$ with $p\le N$ form a set of size at most $\pi(N)$. Thus the total number of integers in $[1,N]$ that can be covered using shifts by elements of $A\cap[1,N]$ is at most $m(N)\pi(N)$ (union bound). Since by hypothesis all $N$ integers are covered, we get $m(N)\pi(N)\ge N$. $\square$

Corollary 2 (Conditional logarithmic lower bound). Assuming the prime number theorem estimate $\pi(N)\sim N/\log N$, Lemma 1 implies \[m(N)\ge (1+o(1))\log N. \]

Proof. From Lemma 1, $m(N)\ge N/\pi(N)$; substitute $\pi(N)\sim N/\log N$. $\square$

5) VERIFICATION - Lemma 1 is a coarse union bound and ignores overlaps; it is still valid. - Corollary 2 depends on the prime number theorem, which is not proved here.

6) FINAL UNRESOLVED

(i) Strongest fully proved partial result: A necessary condition for covering $[1,N]$ is $|A\cap[1,N]|\ge N/\pi(N)$ (Lemma 1).

(ii) First gap: No construction here achieves $|A\cap[1,N]|=o((\log N)^2)$, and no unconditional lower bound of order $\log N$ is proved without using deep estimates for $\pi(N)$.

(iii) Top 3 next moves: 1. Implement Erdős's probabilistic construction (as described in the problem statement) in a self-contained way. 2. Strengthen Lemma 1 by incorporating overlaps (e.g. via inclusion--exclusion or sieve) to approach the constant $e^\gamma$ bound. 3. Investigate whether $O(\log N)$ is possible by building explicit near-optimal complements.

(iv) Minimal counterexample structure (if $O(\log N)$ were impossible): An obstruction showing that for infinitely many $N$, the shifts $\mathcal P + a$ (for $a\in A\cap[1,N]$) cannot cover $[1,N]$ unless $|A\cap[1,N]|$ exceeds $C\log N$ for some $C>0$.