gpt pro 5.4 attempt on Math-5382

Erdős Problem 33 --- Round 3

1. ROUND-3 OBJECTIVE

Path pursued: (A) proof-direction (gap-closure).

The exact gap left after Round 2 was that we had only proved a lower bound for \[ \liminf_{N\to\infty}\frac{W(N)}{\sqrt N}, \] namely the sharp classical barrier $4/\pi$, whereas Erdős Problem 33 asks for the smallest possible value of the limsup \[ L(W):=\limsup_{N\to\infty}\frac{W(N)}{\sqrt N} \] among additive complements $W$ of the squares.

The most promising Round 3 direction is therefore to attack the actual target quantity $L(W)$. The new goal is to show that the natural Round 2 barrier $4/\pi$ cannot in fact be attained by the limsup, and to extract an explicit lower bound for $L(W)$.

2. Round-2 FOUNDATION USED

I use the following Round 2 facts without re-proving them.

1. The basic setup: $W\subseteq \mathbb N$ is an additive complement of the squares, and \[ W(N)=|W\cap\{1,\dots,N\}|. \]

2. The Round 2 theorem: \[ \liminf_{N\to\infty}\frac{W(N)}{\sqrt N}\ge \frac{4}{\pi}. \] This was obtained there by a weighted covering argument.

I will explicitly use this Round 2 result as the established baseline barrier and then go beyond it.

3. NEW INSIGHT / TOOL (ROUND-3)

The new ingredient is an external representation-excess theorem due to Ding and Zhang. Define \[ f_W(n):=\#\{(w,m):n=w+m^2,\; w\in W,\; m\ge 1\}. \] Then Ding--Zhang prove that every additive complement $W$ of the squares satisfies \[ \sum_{n\le N} f_W(n)-N\ge c_0 N \] for all sufficiently large $N$, where \[ c_0:=\frac{1}{4\pi\left(e^{1+2\pi}+1\right)}\approx 5.4631697\cdot 10^{-5}. \]

This has two consequences that were not available in Round 2:

1. it forces a strict improvement over the barrier $4/\pi$ for the actual limsup, and
2. it rules out the entire asymptotic class \[ W(N)=\frac{4}{\pi}\sqrt N+o(\sqrt N). \]

So the Round 3 novelty is not just a better constant: it is a genuine obstruction to any construction trying to saturate the Round 2 barrier on the limsup side.

4. ATTACK PLAN (ROUND-3)

The exact remaining gap after Round 2 was:

Round 2 controlled $\liminf W(N)/\sqrt N$, but the Erdős problem asks for the minimum possible $\limsup W(N)/\sqrt N$.

To close part of that gap, it is enough to prove the following two claims.

1. Claim A. Every additive complement $W$ of the squares satisfies \[ L(W)\ge \frac{4}{\pi}\left(1+\frac{1}{4\pi\left(e^{1+2\pi}+1\right)}\right). \]

2. Claim B. Consequently, no additive complement can satisfy \[ W(N)=\frac{4}{\pi}\sqrt N+o(\sqrt N). \]

The reason this plan works is that an eventual upper bound \[ W(M)\le \alpha\sqrt M \] translates into an upper bound for the representation count \[ \sum_{n\le N} f_W(n)=\sum_{1\le m<\sqrt N} W(N-m^2), \] and the quarter-circle integral turns this into \[ \sum_{n\le N} f_W(n)\lesssim \alpha\frac{\pi}{4}N. \] The Ding--Zhang theorem says the left-hand side is actually at least $(1+c_0)N$, so necessarily \[ \alpha\ge \frac{4}{\pi}(1+c_0). \]

5. WORK (ROUND-3)

Let \[ L(W):=\limsup_{N\to\infty}\frac{W(N)}{\sqrt N} \qquad\text{and}\qquad c_0:=\frac{1}{4\pi\left(e^{1+2\pi}+1\right)}. \]

External Theorem 5.1 (Ding--Zhang).

Let $W$ be an additive complement of the squares and define \[ f_W(n):=\#\{(w,m):n=w+m^2,\; w\in W,\; m\ge 1\}. \] Then for all sufficiently large $N$, \[ \sum_{n\le N} f_W(n)-N\ge c_0 N, \qquad c_0=\frac{1}{4\pi\left(e^{1+2\pi}+1\right)}. \]

Verification of hypotheses. The theorem requires only that $W$ be an additive complement of the squares. This is exactly our standing assumption. No further hypotheses are needed.

Proposition 5.2.

Every additive complement $W$ of the squares satisfies \[ L(W)\ge \frac{4}{\pi}(1+c_0) =\frac{4}{\pi}\left(1+\frac{1}{4\pi\left(e^{1+2\pi}+1\right)}\right). \]

Proof.

Assume for contradiction that \[ L(W)<\frac{4}{\pi}(1+c_0). \] Choose $\eta>0$ such that \[ L(W)+\eta<\frac{4}{\pi}(1+c_0). \] By the definition of limsup, there exists $M_0$ such that for all $M\ge M_0$, \[ W(M)\le (L(W)+\eta)\sqrt M. \]

Now fix $N$ large. By definition of $f_W$, \[ \sum_{n\le N} f_W(n)=\sum_{1\le m<\sqrt N} W(N-m^2). \] Split this sum as \[ \sum_{n\le N} f_W(n)=\Sigma_1+\Sigma_2, \] where \[ \Sigma_1:=\sum_{\substack{1\le m<\sqrt N\\ N-m^2\ge M_0}} W(N-m^2), \qquad \Sigma_2:=\sum_{\substack{1\le m<\sqrt N\\ 0< N-m^2< M_0}} W(N-m^2). \]

For $\Sigma_1$, the eventual upper bound for $W$ gives \[ \Sigma_1\le (L(W)+\eta) \sum_{\substack{1\le m<\sqrt N\\ N-m^2\ge M_0}} \sqrt{N-m^2} \le (L(W)+\eta)\sum_{1\le m<\sqrt N}\sqrt{N-m^2}. \] The function $x\mapsto \sqrt{N-x^2}$ is decreasing on $[0,\sqrt N]$, so \[ \sum_{1\le m<\sqrt N}\sqrt{N-m^2} \le \int_0^{\sqrt N}\sqrt{N-x^2}\,dx =\frac{\pi}{4}N. \] Hence \[ \Sigma_1\le (L(W)+\eta)\frac{\pi}{4}N. \]

For $\Sigma_2$, the integers $N-m^2$ are distinct and lie in $\{1,2,\dots,M_0-1\}$, so there are at most $M_0-1$ such terms. Therefore \[ \Sigma_2\le M_0\,W(M_0)=O(1) \] as $N\to\infty$, with the implied constant depending only on $M_0$ and $W$.

Combining the estimates for $\Sigma_1$ and $\Sigma_2$, we obtain \[ \sum_{n\le N} f_W(n) \le (L(W)+\eta)\frac{\pi}{4}N+O(1). \] Since $L(W)+\eta<\frac{4}{\pi}(1+c_0)$, the coefficient on the right is strictly less than $1+c_0$. Hence for all sufficiently large $N$, \[ \sum_{n\le N} f_W(n)<(1+c_0)N, \] contradicting External Theorem 5.1.

This contradiction proves Proposition 5.2. $\square$

Corollary 5.3.

If \[ LS:=\inf_W L(W) \] denotes the Erdős Problem 33 quantity, where the infimum is over additive complements $W$ of the squares, then \[ LS\ge \frac{4}{\pi}\left(1+\frac{1}{4\pi\left(e^{1+2\pi}+1\right)}\right) \approx 1.273309104. \]

Proof.

Take the infimum of Proposition 5.2 over all admissible $W$. $\square$

Corollary 5.4 (obstruction to the Round-2 barrier).

There is no additive complement $W$ of the squares such that \[ W(N)=\frac{4}{\pi}\sqrt N+o(\sqrt N). \] More generally, one cannot have \[ \limsup_{N\to\infty}\frac{W(N)}{\sqrt N}\le \frac{4}{\pi}. \]

Proof.

Either condition would imply \[ L(W)\le \frac{4}{\pi}, \] which contradicts Proposition 5.2. $\square$

Status update.

Combining Corollary 5.3 with van Doorn's construction as recorded on the Erdős Problems site, the currently verified interval for Problem 33 is \[ 1.273309104\ldots \le LS \le 2\varphi^{5/2} \approx 6.660381354, \qquad \varphi=\frac{1+\sqrt5}{2}. \]

6. ADVERSARIAL VERIFICATION

1. Was the external theorem applied with the correct hypotheses? Yes. External Theorem 5.1 requires only that $W$ be an additive complement of the squares, which is exactly our standing assumption.

2. Is the limsup-to-eventual-bound step valid? Yes. From \[ L(W)=\limsup_{N\to\infty} W(N)/\sqrt N \] and any $\eta>0$, it follows that $W(M)\le (L(W)+\eta)\sqrt M$ for all sufficiently large $M$.

3. Is there a hidden uniformity gap for small values of $N-m^2$? No. Those terms were separated into $\Sigma_2$, and their total contribution is $O(1)$, independent of $N$.

4. Is the quarter-circle step correct? Yes, but it needs an inequality rather than an equality: \[ \sum_{1\le m<\sqrt N}\sqrt{N-m^2} \le \int_0^{\sqrt N}\sqrt{N-x^2}\,dx =\frac{\pi}{4}N. \] This fixes the compressed step that is often written informally in short proofs.

5. Does this really advance beyond Round 2? Yes. Round 2 gave only \[ \liminf W(N)/\sqrt N\ge 4/\pi. \] Round 3 proves a nontrivial lower bound for the actual Erdős objective $L(W)=\limsup W(N)/\sqrt N$, and it rules out the entire natural barrier class $W(N)=\frac{4}{\pi}\sqrt N+o(\sqrt N)$.

7. FINAL (EXACTLY ONE)

UNRESOLVED (BUT STRICTLY ADVANCED).

Round 3 gives a genuine new theorem about the actual target quantity in Erdős Problem 33: \[ \inf_W \limsup_{N\to\infty}\frac{W(N)}{\sqrt N} \ge \frac{4}{\pi}\left(1+\frac{1}{4\pi\left(e^{1+2\pi}+1\right)}\right) \approx 1.273309104. \] In particular, the natural Round 2 barrier $4/\pi$ cannot be attained by the limsup, and any proof strategy or construction aiming for \[ W(N)=\frac{4}{\pi}\sqrt N+o(\sqrt N) \] is now rigorously ruled out.

8. COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 49%

9. REFERENCES

1. Y. Ding and Z. Zhang, No exact on average additive complements of squares, arXiv:2512.15407, 2025. Used for External Theorem 5.1.
2. T. F. Bloom, Erdős Problem #33, https://www.erdosproblems.com/33, accessed 2026-03-07. Used for the problem statement and the currently recorded van Doorn upper bound.

10. OUTPUT FORMAT

Detected Erdős problem number: $33$.

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