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Math-655-A4 · revision r1 · on Math-655

gpt pro 5.2 attempt on Math-655, version 2

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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. Erdős Problem #41 (Round 3)
  2. Problem
  3. 1) ROUND-3 OBJECTIVE
  4. 2) Round-2 foundation used
  5. 3) New insight / tool (Round 3)
  6. 4) Attack plan (Round 3)
  7. 5) Work (Round 3)
  8. 5.1. Local $O(N^{1/3})$ bound on every interval.
  9. 5.2. What Helm actually proves (and what he does not).
  10. 5.3. Correct partial consequence of Helm + Lemma 41.1.
  11. 5.4. Helm's gap-block theorem (what any counterexample must look like).
  12. 5.5. New lemma: early gap blocks force small normalized density.
  13. 6) Adversarial verification
  14. 7) FINAL
  15. 8) Completion estimate
  16. 9) References

Erdős Problem #41 (Round 3)

Problem

An infinite set $A\subset\mathbb N$ is a $B_3$-set if all sums \[ a_1+a_2+a_3\qquad (a_1\le a_2\le a_3,\ a_i\in A) \] are distinct. The Erdős question is whether every infinite $B_3$-set satisfies \[ \liminf_{N\to\infty}\frac{|A\cap[1,N]|}{N^{1/3}}=0. \]

1) ROUND-3 OBJECTIVE

Path (C): obstruction/correction + strict progress. Round 2 claimed a full proof by invoking "Helm obstructions" $H1$--$H2$ as if they were unconditional. In this round we (i) identify the precise missing hypothesis in Helm's Theorem 1 (the proof only works for monotonically distributed sequences), (ii) record the strongest correct consequence obtainable from Helm (pseudo-cubes and monotone distribution are ruled out), and (iii) add a new structural lemma (a local version of Lemma 41.1) and a quantitative "early-gap  $\Rightarrow$ small ratio" criterion. This isolates a concrete remaining obstruction: to prove Erdős' liminf statement it would suffice to force Helm's long gap blocks to begin sufficiently early.

2) Round-2 foundation used

We rely on the following vetted Round 1 facts.
  • Lemma 41.1 (Round 1). If $A\subseteq[1,N]$ is $B_3$ and $m:=|A|$, then $\binom{m+2}{3}\le 3N-2$, hence $m\le (18N)^{1/3}+2$.
  • Lemma 41.2 (Round 1). Every infinite $B_3$-set is Sidon ($B_2$).

3) New insight / tool (Round 3)

  1. Correct reading of Helm $1996$. Helm's key non-$B_3$ criterion (his Theorem 1) assumes the sequence is monotonically distributed. This hypothesis was omitted in Round 2, so the Round 2 "full proof" does not go through.
  2. Helm's Theorem 3 (gap blocks). Under the (still hypothetical) assumption $\liminf A(n)n^{-1/3}>0$, Helm proves the existence of long consecutive blocks of unusually sparse length-$N$ intervals ("$\omega$-gaps").
  3. Strengthening Lemma 41.1 to a local interval bound. Any translate of a finite $B_3$ set is $B_3$, hence the $O(N^{1/3})$ bound holds in every interval of length $N$.
  4. New quantitative criterion. If Helm-type gap blocks occur early enough relative to their length, then the Erdős liminf conclusion follows.

4) Attack plan (Round 3)

  • Gap after Round 2. Round 2 treated Helm's Theorem 1 as an unconditional obstruction $H2$. In fact Helm assumes monotone distribution. There is no proof (and it is not true in general) that every $B_3$ set is monotonically distributed.
  • What we can still do. We state the correct Helm theorems and derive rigorous partial consequences. We then combine them with the local interval bound to obtain a clean "if the gap block starts early, then the normalized counting function dips to 0" lemma.
  • Why this is progress. It isolates one sharply formulated missing step: show that Helm's gap blocks (which exist at all large scales under $\liminf>0$) can be taken to start within $o(\log^{1/6}N)$ blocks for infinitely many scales. That condition would settle Erdős Problem 41.

5) Work (Round 3)

5.1. Local $O(N^{1/3})$ bound on every interval.
Lemma (Local version of Lemma 41.1): Let $A\subseteq\mathbb N$ be a $B_3$ set. For any integers $M\ge 0$ and $N\ge 1$, \[ |A\cap (M,M+N]|\le (18N)^{1/3}+2. \]
Proof: Let $B:=\{a-M: a\in A\cap(M,M+N]\}$. Then $B\subseteq[1,N]$ and $B$ is $B_3$ (translation preserves triple-sum distinctness). Apply Lemma 41.1 (Round 1) to $B$. ∎
5.2. What Helm actually proves (and what he does not).
We record the parts of Helm's paper that are directly relevant.
Definition (Helm: monotonically distributed): Let $A$ be an increasing sequence of naturals and $A(x):=|A\cap[1,x]|$. For $N\in\mathbb N$ and $l\in\{1,\dots,N\}$ set \[ I_l^{(N)}:=((l-1)N,\,lN],\qquad A_l^{(N)}:=|A\cap I_l^{(N)}|. \] We call $A$ monotonically distributed if there exist functions $\psi:\mathbb N\to\mathbb N$ with $\psi(N)\to\infty$ and a sequence $N_r\to\infty$ such that for all $r$ and all $1\le l<\psi(N_r)$, \[ A_l^{(N_r)}\ge A_{l+1}^{(N_r)}. \]
Theorem (Helm, Theorem 1 (precise)): If $A$ is monotonically distributed and \[ 0<\liminf_{n\to\infty}\frac{A(n)}{n^{1/3}}<\limsup_{n\to\infty}\frac{A(n)}{n^{1/3}}<\infty, \] then $A$ is not a $B_3$-sequence.
Theorem (Helm, Theorem 2 (pseudo-cubes)): If $A(n)\sim \alpha n^{1/3}$ for some $\alpha>0$ ("pseudo-cube"), then $A$ is not a $B_3$-sequence.
Remark (Correction of the Round 2 "closing step"): Round 2 asserted two unconditional obstructions:
$H1$ $A(n)\sim\alpha n^{1/3}$ implies not $B_3$;  $H2$ $0<\liminf A(n)n^{-1/3}<\limsup A(n)n^{-1/3}<\infty$ implies not $B_3$.
$H1$ matches Helm's Theorem 2. However $H2$ is missing the hypothesis "$A$ is monotonically distributed" from Helm's Theorem 1. Thus the Round 2 deduction "$\liminf>0$ forces a contradiction" is not valid.
5.3. Correct partial consequence of Helm + Lemma 41.1.
Corollary (Corrected Round 2 conclusion): Let $A\subset\mathbb N$ be an infinite $B_3$-set and $A(n)=|A\cap[1,n]|$.
  1. If $A$ is monotonically distributed, then necessarily \[ \liminf_{n\to\infty}\frac{A(n)}{n^{1/3}}=0. \]
  2. In particular, no $B_3$-sequence can be pseudo-cube ($A(n)\sim \alpha n^{1/3}$).
Proof: By Lemma 41.1 (Round 1), the normalized ratio $A(n)/n^{1/3}$ is bounded above. If $A$ were monotonically distributed and had $\liminf>0$, then either $\liminf<\limsup$ and Helm Theorem Theorem (Helm, Theorem 1 (precise)) contradicts $B_3$, or $\liminf=\limsup>0$ and Helm Theorem Theorem (Helm, Theorem 2 (pseudo-cubes)) contradicts $B_3$. ∎
5.4. Helm's gap-block theorem (what any counterexample must look like).
Definition ($\omega$-gap (Helm)): Given $\omega:\mathbb N\to(0,\infty)$ and a scale $N$, we call $I_l^{(N)}$ an $\omega$-gap if \[ A_l^{(N)}<\frac{N^{1/3}}{\omega(N)\,l^{2/3}}. \]
Theorem (Helm, Theorem 3): Assume $A$ is a $B_3$-sequence with $\liminf_{n\to\infty}A(n)n^{-1/3}>0$. Fix $\delta>0$ and define $\omega(n):=(\log n)^{1/6-\delta}$. Then for any sufficiently large $N$ and any $\alpha$ with $0<\alpha\le 1/2$, there exist at least \[ \kappa_N:=\lfloor \log^{1/6}N\rfloor \] consecutive $\omega$-gaps of $A$ below $N^{1+\alpha}$. Equivalently, there exists $l_0<N^\alpha$ such that for all $l_0<l\le l_0+\kappa_N$, \[ A_l^{(N)}<\frac{N^{1/3}}{l^{2/3}(\log N)^{1/6-\delta}}. \]
Remark (Interpretation): Theorem Theorem (Helm, Theorem 3) shows that any hypothetical $B_3$ counterexample to Erdős' liminf claim must be extremely non-uniform at every large scale: for every large $N$ it contains a block of $\asymp\log^{1/6}N$ consecutive length-$N$ intervals each having an extra $(\log N)^{1/6-\delta}$ factor of sparsity beyond the "natural" $l^{-2/3}$ decay.
5.5. New lemma: early gap blocks force small normalized density.
Theorem Theorem (Helm, Theorem 3) does not control where the gap block begins. The next lemma shows that if the gap block begins early enough (relative to its length), then Erdős' liminf conclusion follows.
Lemma (Early consecutive gaps $\Rightarrow$ small ratio): Let $A\subseteq\mathbb N$ be a $B_3$-set and write $A(x)=|A\cap[1,x]|$. Fix $N\ge 1$ and integers $l_0\ge 0$ and $K\ge 1$. Assume that for each $l=l_0+1,\dots,l_0+K$ we have \begin{equation} \label{wsref-0-65-71-3a-67-61-70-2d-68-79-70} A_l^{(N)}\le \frac{N^{1/3}}{\omega(N)\,l^{2/3}} \end{equation} for some parameter $\omega(N)\ge 1$. Then, with $X:=(l_0+K)N$, we have \begin{equation} \label{wsref-0-65-71-3a-72-61-74-69-6f-2d-62-6f-75-6e-64} \frac{A(X)}{X^{1/3}} \le (18)^{1/3}\Big(\frac{l_0}{l_0+K}\Big)^{1/3} +\frac{C}{\omega(N)}\Big(\frac{K}{l_0+K}\Big)^{1/3}, \end{equation} where $C>0$ is an absolute constant. In particular, if $\omega(N)\to\infty$ along a sequence of $N$ and $l_0=o(K)$ along that same sequence, then \[ \liminf_{x\to\infty}\frac{A(x)}{x^{1/3}}=0. \]
Proof: Write $X=(l_0+K)N$. By monotonicity of $A(\cdot)$, \[ A(X)=A(l_0N)+\sum_{l=l_0+1}^{l_0+K}A_l^{(N)}. \] First, Lemma 41.1 (Round 1) applied to the initial segment $[1,l_0N]$ gives $A(l_0N)\le (18l_0N)^{1/3}+2$. Second, the gap hypothesis (eq:gap-hyp) gives \[ \sum_{l=l_0+1}^{l_0+K}A_l^{(N)} \le \frac{N^{1/3}}{\omega(N)}\sum_{l=l_0+1}^{l_0+K}\frac{1}{l^{2/3}}. \] The sum satisfies $\sum_{l=l_0+1}^{l_0+K}l^{-2/3}\ll \int_{l_0}^{l_0+K}t^{-2/3}\,dt\ll (l_0+K)^{1/3}-l_0^{1/3}\ll K^{1/3}$, hence \[ \sum_{l=l_0+1}^{l_0+K}A_l^{(N)}\ll \frac{N^{1/3}K^{1/3}}{\omega(N)}. \] Divide by $X^{1/3}=N^{1/3}(l_0+K)^{1/3}$ and absorb the additive "$+2$" into the constant to obtain (eq:ratio-bound). If $l_0=o(K)$ and $\omega(N)\to\infty$, the right-hand side tends to $0$. ∎
Remark (A concrete reduction): Combine Helm's Theorem Theorem (Helm, Theorem 3) with Lemma lem:early-gap. If one could strengthen Helm's conclusion to ensure that the block of $\kappa_N\asymp\log^{1/6}N$ consecutive $\omega$-gaps may be chosen with start index $l_0=o(\kappa_N)$ infinitely often, then Erdős' liminf conjecture would follow.

6) Adversarial verification

  • Translation invariance (Lemma Lemma (Local version of Lemma 41.1)). Triple-sum distinctness is preserved under $a\mapsto a-M$ since $a+b+c=a'+b'+c'$ iff $(a-M)+(b-M)+(c-M)=(a'-M)+(b'-M)+(c'-M)$. Hence the local bound is rigorous.
  • The Round 2 flaw. Helm Theorem 1 explicitly assumes "monotonically distributed". Dropping this hypothesis changes the statement and is not justified by any argument in Round 2.
  • Lemma lem:early-gap constants. The proof only uses (i) Lemma 41.1 (Round 1) at $l_0N$, and (ii) the integral comparison $\sum_{l=l_0+1}^{l_0+K}l^{-2/3}\ll K^{1/3}$. No hidden hypotheses.
  • Edge cases. If $l_0=0$, then the first term in (eq:ratio-bound) vanishes and we get $A(KN)/(KN)^{1/3}\ll 1/\omega(N)$, consistent with the formula.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED).
  • The Round 2 "full proof" is not correct: it relies on an unconditional form of Helm's Theorem 1 that omits the needed monotonic distribution hypothesis.
  • What is proved: Erdős' liminf statement holds for the important regularity classes covered by Helm (monotonically distributed sequences and pseudo-cubes), and any putative counterexample with $\liminf>0$ must exhibit Helm's long consecutive gap blocks at every scale.
  • New progress: Lemma Lemma (Local version of Lemma 41.1) (local interval bound) and Lemma lem:early-gap (early gap blocks force the normalized counting function to dip to $0$) isolate a concrete missing step that would settle the conjecture.

8) Completion estimate

COMPLETION: 65%

9) References

  • M. Helm, On the distribution of $B_3$-sequences, J. Number Theory 58 $1996$, 124--129.
  • (For current-status bookkeeping) Erdős Problems website entry for Problem #41.