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Math-649-A5 · revision r1 · on Math-649

gpt pro 5.2 attempt on Math-649, version 3

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Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Round 3 (Problem 30): Sidon sets in $\{1,\dots,N\}$
  2. 1) Round-3 Objective
  3. 2) Round-2 Foundation Used
  4. 3) New Insight / Tool (Round 3)
  5. 4) Attack Plan (Round 3)
  6. 5) Work (Round 3): record upper bound via diameter
  7. 6) Adversarial Verification
  8. 7) Final
  9. 8) Completion Estimate
  10. 9) References

Round 3 (Problem 30): Sidon sets in $\{1,\dots,N\}$

1) Round-3 Objective

Path (A): proof direction (sharpen the best-known upper bound). Round 2 established the classical Erdős--Turán/Lindström bound \[ h(N)\le \sqrt N+N^{1/4}+1. \] In this round we strictly strengthen the upper bound, importing the current record on the $N^{1/4}$-term and explaining how it implies an improved estimate for $h(N)$: \[ h(N)\le \sqrt N+0.98183\,N^{1/4}+O(1). \] The original Erdős--Turán question asking whether $h(N)=\sqrt N+O_\varepsilon(N^\varepsilon)$ for every $\varepsilon>0$ remains open.

2) Round-2 Foundation Used

We use the following vetted Round-2 results/notation.
  • (R2 Theorem 30.4) (Lindström bound) For every $N$, \[ h(N)\le \sqrt N+N^{1/4}+1. \]
  • (R1 Lemma 30.1) Sidon $\Rightarrow$ all positive differences are distinct.

We do not re-prove these statements; they serve as the baseline to be improved.

3) New Insight / Tool (Round 3)

Diameter viewpoint + best-known diameter lower bounds. For a finite set $A\subset \mathbb Z$, define the diameter \[ \mathrm{diam}(A):=\max A-\min A. \] If $A\subseteq \{1,\dots,N\}$ then $\mathrm{diam}(A)\le N-1$. Recent work of Carter--Hunter--O'Bryant provides an improved lower bound on $\mathrm{diam}(A)$ in terms of $|A|$ for Sidon sets, which is equivalent to an improved upper bound on $h(N)$ with a better constant in front of $N^{1/4}$.

4) Attack Plan (Round 3)

Remaining gap after Round 2: the exponent $1/4$ in the error term is far from the conjectural $O_\varepsilon(N^\varepsilon)$.

Round-3 goal: improve the constant in the known $N^{1/4}$ upper bound. Concretely:

  1. import the current-record upper bound $h(N)\le \sqrt N + 0.98183\,N^{1/4}+O(1)$,
  2. check that it applies to the present integer-interval model $A\subseteq\{1,\dots,N\}$,
  3. record the (strictly stronger) state of the art as a new theorem/corollary beyond Round 2.

5) Work (Round 3): record upper bound via diameter

Theorem (Carter--Hunter--O'Bryant (record bound)): There exists an absolute constant $C$ such that for all sufficiently large integers $n$, every Sidon set $A\subset \mathbb Z$ with $\mathrm{diam}(A)\le n$ satisfies \[ |A|\le n^{1/2}+0.98183\,n^{1/4}+C. \] Equivalently, if $|A|=k$ then \[ \mathrm{diam}(A)\ \ge\ k^2-b\,k^{3/2}-O(k)\qquad\text{with }b\le 1.96365. \] Moreover, the same paper gives a hand-checkable bound with $b\le 1.99058$ (hence a slightly weaker constant in front of $n^{1/4}$).
Proof[Justification / source]: This is Theorem (equivalent forms) as stated in the abstract of Carter--Hunter--O'Bryant, On the Diameter of Finite Sidon Sets (arXiv:2310.20032), which explicitly records both the constant $0.98183$ and the corresponding $b\le 1.96365$, as well as the hand-verifiable $b\le 1.99058$. ∎
Corollary (Improved upper bound for $h(N)$): There exists an absolute constant $C'$ such that for all sufficiently large $N$, \[ h(N)\le \sqrt N+0.98183\,N^{1/4}+C'. \] In particular, Round 2's coefficient $1$ in front of $N^{1/4}$ can be replaced by $0.98183$ (with an $O(1)$ loss).
Proof: Let $A\subseteq \{1,\dots,N\}$ be Sidon with $|A|=h(N)$. Then $\mathrm{diam}(A)\le N-1$. Apply Theorem Theorem (Carter--Hunter--O'Bryant (record bound)) with $n:=N-1$ to obtain, for sufficiently large $N$, \[ h(N)=|A|\le (N-1)^{1/2}+0.98183\,(N-1)^{1/4}+C. \] Since $(N-1)^{1/2}\le N^{1/2}$ and $(N-1)^{1/4}\le N^{1/4}$, the right-hand side is \[ \le \sqrt N+0.98183\,N^{1/4}+C, \] which proves the claim with $C':=C$. ∎
Remark (State of the art (as of Oct. 2025 / Jan. 2026)): The Erdős Problems website for Problem #30 records the progression \[ \sqrt N+N^{1/4}+1\ \to\ \sqrt N+0.998\,N^{1/4}\ \to\ \sqrt N+0.99703\,N^{1/4}\ \to\ \sqrt N+0.98183\,N^{1/4}+O(1), \] with the current record attributed to Carter--Hunter--O'Bryant.

6) Adversarial Verification

We check that no hidden assumptions were imported incorrectly.
  • Model compatibility. Theorem Theorem (Carter--Hunter--O'Bryant (record bound)) is stated for integer Sidon sets with bounded diameter; any $A\subseteq\{1,\dots,N\}$ satisfies $\mathrm{diam}(A)\le N-1$, so the hypothesis holds with $n=N-1$.
  • Quantifiers. Theorem Theorem (Carter--Hunter--O'Bryant (record bound)) holds for all sufficiently large $n$, hence Corollary cor:hN-record holds for all sufficiently large $N$; this is consistent with the $O(1)$ term.
  • Off-by-one in $n=N-1$. Replacing $N-1$ by $N$ only weakens the bound since $x^{1/2}$ and $x^{1/4}$ are increasing.
  • Interaction with Round 2. Round 2 provided a self-contained coefficient-$1$ bound; Round 3 strictly improves the coefficient using an external theorem. No step contradicts Round 2.

7) Final

UNRESOLVED (BUT STRICTLY ADVANCED). Round 3 improves the Round-2 upper bound by importing the current-record estimate \[ h(N)\le \sqrt N+0.98183\,N^{1/4}+O(1), \] which is strictly stronger than $\sqrt N+N^{1/4}+1$ for large $N$. The main open target $h(N)=\sqrt N+O_\varepsilon(N^\varepsilon)$ remains unsolved.

8) Completion Estimate

COMPLETION: 70%.

9) References

  • T. F. Bloom (ed.), Erdős Problem #30, https://www.erdosproblems.com/30 (page last edited 19 Oct 2025; accessed 20 Jan 2026).
  • D. Carter, Z. Hunter, K. O'Bryant, On the Diameter of Finite Sidon Sets, arXiv:2310.20032 (also Acta Math. Hungar. 175$1$ $2025$, 108--126).
  • J. Balogh, Z. Füredi, S. Roy, An upper bound on the size of Sidon sets, arXiv:2103.15850 $2021$.
  • K. O'Bryant, On the size of finite Sidon sets, arXiv:2207.07800 (journal ref. Ukrains'kyi Matematychnyi Zhurnal, 76$8$ (Sept 2024)).