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Math-5388-A10 · revision r1 · on Math-5388

gpt pro 5.4 attempt on Math-5388, version 5

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

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Contents
  1. Erdős Problem #43 (Round 8)
  2. 1) ROUND-8 OBJECTIVE
  3. 2) Round-7 FOUNDATION USED
  4. 3) NEW INSIGHT / TOOL (ROUND-8)
  5. (i) Fejér--Riesz spectral factorisation closes the last loophole in the positive-kernel method.
  6. (ii) Exact optimisation of the whole positive-definite-weight framework.
  7. 4) ATTACK PLAN (ROUND-8)
  8. 5) WORK (ROUND-8)
  9. 5.1 Admissible weights and Fejér--Riesz factorisation
  10. 5.2 The universal positive-definite-kernel inequality
  11. 5.3 Fejér's inequality and the unique optimiser
  12. 5.4 Consequences and relation to Round 7
  13. 6) ADVERSARIAL VERIFICATION
  14. (i) Did Round 8 really strengthen Round 7?
  15. (ii) Hidden hypothesis in Fejér--Riesz.
  16. (iii) Why is $W=\sum_n w(n)$ nonzero?
  17. (iv) Equality case in Lemma Lemma (Fejér inequality in coefficient form).
  18. (v) Does this settle the first question?
  19. 7) FINAL
  20. 8) COMPLETION ESTIMATE (MANDATORY)
  21. 9) REFERENCES

Erdős Problem #43 (Round 8)

1) ROUND-8 OBJECTIVE

Path chosen: (C) --- obstruction/correction for the remaining open part.

Round 7 ruled out the entire finite positive-combination version of the Tao-style kernel method, but it still left one genuine loophole: perhaps one could beat the Round 7 barrier by using a single general nonnegative positive-definite weight that is not presented in advance as a nonnegative-generator autocorrelation, or by hiding such a weight inside a more elaborate kernel package.

The most promising Round 8 direction is therefore to close that loophole completely. I prove a universal positive-definite-kernel barrier: among all finitely supported nonnegative positive-definite weights, the unique optimal choice is the discrete Fejér kernel, and the best bound obtainable is exactly the same interval-kernel bound already visible in Rounds 6--7. This strictly strengthens Round 7 by removing both the "nonnegative generator" and "finite package" restrictions.

2) Round-7 FOUNDATION USED

I explicitly use the following Round 7 results.

  • Round 7 pointwise input. If $A,B\subset[N]$ are Sidon and $(A-A)\cap(B-B)=\{0\}$, then, writing \[ \Delta_A:=\mathbf 1_A*\mathbf 1_{-A},\qquad \Delta_B:=\mathbf 1_B*\mathbf 1_{-B}, \] one has \[ \Delta_A(n)+\Delta_B(n)\le (|A|+|B|)\mathbf 1_{\{0\}}(n)+1 \qquad (n\in\mathbb Z). \]
  • Round 7 finite positive-combination barrier. Any finite nonnegative linear combination of interval-kernel inequalities is no stronger than the best single interval kernel.
  • Round 4 asymptotic lower family. There exist infinitely many feasible pairs with \[ |A|+|B|=(\sqrt 2+o(1))\sqrt N. \] This will be used to quantify the strength of the new barrier.

3) NEW INSIGHT / TOOL (ROUND-8)

There are two genuinely new ingredients in this round.

(i) Fejér--Riesz spectral factorisation closes the last loophole in the positive-kernel method.
I work with an arbitrary finitely supported weight $w:\mathbb Z\to [0,\infty)$ whose Fourier transform \[ \widehat w(\theta):=\sum_{n\in\mathbb Z} w(n)e^{2\pi i n\theta} \] is nonnegative for all $\theta\in\mathbb R/\mathbb Z$ (equivalently, $w$ is positive-definite). By the classical Fejér--Riesz theorem, such a $w$ factors as \[ w=\psi*\widetilde\psi \] for some finitely supported complex $\psi$. This lets me re-run the Round 6/7 argument for the full class of nonnegative positive-definite weights, not just the special weights built from nonnegative generators.
(ii) Exact optimisation of the whole positive-definite-weight framework.
If $w$ is supported in $\{-(L-1),\dots,L-1\}$, the best upper bound obtainable from $w$ is \[ B_w(N,T):=(N+L-1)\left(1+T\frac{w(0)}{\sum_n w(n)}\right), \qquad T:=|A|+|B|. \] I prove \[ \sum_n w(n)\le L\,w(0), \] with equality if and only if $w$ is a positive scalar multiple of the discrete Fejér kernel \[ F_L(n):=\max\{L-|n|,0\}. \] Hence the exact optimum over all admissible positive-definite weights is \[ \mathcal B_{\mathrm{pd}}(N,T)=\min_{L\ge 1}(N+L-1)\left(1+\frac{T}{L}\right), \] and it is uniquely attained (up to scale) by the Fejér kernels. This strictly strengthens Round 7.

4) ATTACK PLAN (ROUND-8)

Exact gap left after Round 7. Round 7 exhausted finite positive combinations of the previous interval/autocorrelation kernels, but not the larger class of all nonnegative positive-definite weights. In principle, such a weight could arise from a sign-changing or complex generator after spectral factorisation, so Round 7 did not yet fully rule out the broader Tao-style framework.

Claims proved in this round.

  1. Every admissible positive-definite weight $w$ yields an upper bound \[ S:=|A|^2+|B|^2\le B_w(N,T):=(N+L-1)\left(1+T\frac{w(0)}{\sum_n w(n)}\right), \] where $L$ is the support-length parameter of $w$.
  2. For every such $w$, one has \[ \sum_n w(n)\le L\,w(0), \] with equality exactly for positive multiples of the Fejér kernel.
  3. Consequently, \[ \inf_w B_w(N,T)=\min_{L\ge 1}(N+L-1)\left(1+\frac{T}{L}\right), \] so the whole positive-definite-weight method is exactly optimised by the same Fejér/interval kernel already present in Round 6.

This overcomes the final loophole in the positive-kernel approach and rules out an even broader class of proof strategies than Round 7.

5) WORK (ROUND-8)

5.1 Admissible weights and Fejér--Riesz factorisation

Let $A,B\subset[N]$ be Sidon sets with $(A-A)\cap(B-B)=\{0\}$, and write \[ a:=|A|,\qquad b:=|B|,\qquad S:=a^2+b^2,\qquad T:=a+b. \]

Call a function $w:\mathbb Z\to[0,\infty)$ admissible of span $L$ if:

  • $w$ is supported in $\{-(L-1),\dots,L-1\}$,
  • $\widehat w(\theta)\ge 0$ for all $\theta\in\mathbb R/\mathbb Z$.
The second condition is equivalent to positive-definiteness of $w$.

We will use the following classical theorem.

Theorem (Fejér--Riesz): Let \[ p(\theta)=\sum_{n=-(L-1)}^{L-1} c_n e^{2\pi i n\theta} \] be a trigonometric polynomial that is nonnegative for all $\theta\in\mathbb R/\mathbb Z$. Then there exists a polynomial \[ P(z)=\sum_{j=0}^{L-1}\psi(j)z^j \] of degree at most $L-1$ such that \[ p(\theta)=\left|P(e^{2\pi i\theta})\right|^2. \] Equivalently, the coefficients satisfy \[ c_n=\sum_{j\in\mathbb Z}\psi(j+n)\overline{\psi(j)}. \]

Applied to $p=\widehat w$, Theorem Theorem (Fejér--Riesz) gives the following factorisation.

Corollary: If $w$ is admissible of span $L$, then there exists a complex-valued function $\psi$ supported in $\{0,1,\dots,L-1\}$ such that \[ w=\psi*\widetilde\psi, \qquad \widetilde\psi(n):=\overline{\psi(-n)}. \]
Proof: Write \[ \widehat w(\theta)=\sum_{n=-(L-1)}^{L-1} w(n)e^{2\pi i n\theta}. \] Since $w$ is admissible, $\widehat w(\theta)\ge 0$ for all $\theta$. Applying Theorem Theorem (Fejér--Riesz) gives \[ \widehat w(\theta)=\left|\sum_{j=0}^{L-1}\psi(j)e^{2\pi i j\theta}\right|^2. \] Comparing Fourier coefficients yields $w=\psi*\widetilde\psi$. ∎

5.2 The universal positive-definite-kernel inequality

Proposition: Let $w$ be an admissible weight of span $L$, and set \[ W:=\sum_{n\in\mathbb Z} w(n). \] Then \begin{equation} \label{wsref-0-65-71-3a-70-64-2d-6d-61-73-74-65-72} S\le (N+L-1)\left(1+T\frac{w(0)}{W}\right). \end{equation}
Proof: By Corollary Corollary, write $w=\psi*\widetilde\psi$ with $\operatorname{supp}(\psi)\subseteq\{0,\dots,L-1\}$.

First, using the Round 7 pointwise estimate, \[ \Delta_A(n)+\Delta_B(n)\le T\mathbf 1_{\{0\}}(n)+1, \] and the fact that $w(n)\ge0$, we obtain \begin{align} \sum_n w(n)\Delta_A(n)+\sum_n w(n)\Delta_B(n) &\le T w(0)+\sum_n w(n)\notag\\ &=T w(0)+W. \label{wsref-0-65-71-3a-75-70-70-65-72-70-64} \end{align}

On the other hand, \[ \sum_n w(n)\Delta_A(n)=\sum_n (\psi*\widetilde\psi)(n)(\mathbf 1_A*\mathbf 1_{-A})(n)=\|\mathbf 1_A*\psi\|_2^2, \] and similarly \[ \sum_n w(n)\Delta_B(n)=\|\mathbf 1_B*\psi\|_2^2. \] Now $\mathbf 1_A*\psi$ is supported in an interval of length at most $N+L-1$, so Cauchy--Schwarz gives \[ \|\mathbf 1_A*\psi\|_2^2\ge \frac{\left|\sum_x (\mathbf 1_A*\psi)(x)\right|^2}{N+L-1} =\frac{a^2\left|\sum_j \psi(j)\right|^2}{N+L-1}. \] Likewise, \[ \|\mathbf 1_B*\psi\|_2^2\ge \frac{b^2\left|\sum_j \psi(j)\right|^2}{N+L-1}. \] Adding and using \[ W=\sum_n w(n)=\left|\sum_j \psi(j)\right|^2, \] we obtain \begin{equation} \label{wsref-0-65-71-3a-6c-6f-77-65-72-70-64} \sum_n w(n)\Delta_A(n)+\sum_n w(n)\Delta_B(n) \ge \frac{SW}{N+L-1}. \end{equation} Combining (eq:upperpd) and (eq:lowerpd) yields (eq:pd-master). ∎

5.3 Fejér's inequality and the unique optimiser

Lemma (Fejér inequality in coefficient form): If $w$ is admissible of span $L$, then \begin{equation} \label{wsref-0-65-71-3a-66-65-6a-65-72-2d-63-6f-65-66-66} W\le L\,w(0). \end{equation} Moreover, equality holds if and only if \[ w(n)=c\,F_L(n) \qquad (n\in\mathbb Z) \] for some $c>0$, where \[ F_L(n):=\max\{L-|n|,0\} \] is the discrete Fejér kernel.
Proof: By Corollary Corollary, write $w=\psi*\widetilde\psi$ with $\operatorname{supp}(\psi)\subseteq\{0,\dots,L-1\}$. Then \[ W=\left|\sum_{j=0}^{L-1}\psi(j)\right|^2, \qquad w(0)=\sum_{j=0}^{L-1}|\psi(j)|^2. \] Applying Cauchy--Schwarz to the length-$L$ vector $(\psi(0),\dots,\psi(L-1))$ gives \[ \left|\sum_{j=0}^{L-1}\psi(j)\right|^2\le L\sum_{j=0}^{L-1}|\psi(j)|^2, \] which is exactly (eq:fejer-coeff).

Equality in Cauchy--Schwarz holds if and only if all $\psi(j)$ have the same phase and the same modulus. Thus, after multiplying by a unimodular constant, one has \[ \psi(j)=c^{1/2} \qquad (0\le j\le L-1) \] for some $c>0$. In that case, \[ w(n)=\sum_j \psi(j+n)\overline{\psi(j)}=c\,\#\{j:0\le j\le L-1,\ 0\le j+n\le L-1\}=c\,F_L(n). \] Conversely, $w=cF_L$ is generated by the constant vector $\psi\equiv c^{1/2}$ on $\{0,\dots,L-1\}$, so equality holds. ∎

Theorem (Universal positive-definite-kernel barrier): Let $\mathcal B_{\mathrm{pd}}(N,T)$ denote the strongest upper bound on $S=|A|^2+|B|^2$ obtainable from the Round 7 pointwise estimate by summing against an arbitrary admissible positive-definite weight and using only the Cauchy--Schwarz lower bound above. Then \begin{equation} \label{wsref-0-65-71-3a-42-70-64} \mathcal B_{\mathrm{pd}}(N,T)=\min_{L\ge 1}(N+L-1)\left(1+\frac{T}{L}\right). \end{equation} For each fixed $L$, the unique optimiser (up to positive scalar multiple) is the discrete Fejér kernel $F_L$.
Proof: Fix an admissible weight $w$ of span $L$. Proposition Proposition gives \[ S\le (N+L-1)\left(1+T\frac{w(0)}{W}\right). \] By Lemma Lemma (Fejér inequality in coefficient form), \[ \frac{w(0)}{W}\ge \frac1L, \] so every admissible weight of span $L$ yields a bound at least \[ (N+L-1)\left(1+\frac{T}{L}\right). \] Conversely, taking $w=F_L=\mathbf 1_{[0,L-1]}*\mathbf 1_{-[0,L-1]}$ gives \[ W=L^2, \qquad w(0)=L, \] so Proposition Proposition yields exactly \[ S\le (N+L-1)\left(1+\frac{T}{L}\right). \] Minimising over $L\ge 1$ proves (eq:Bpd). The uniqueness statement for fixed $L$ follows from the equality case in Lemma Lemma (Fejér inequality in coefficient form). ∎

5.4 Consequences and relation to Round 7

Corollary: Round 7's finite positive-combination barrier is a special case of Theorem Theorem (Universal positive-definite-kernel barrier).
Proof: A finite nonnegative combination of Round 7 kernels has the form \[ w=\sum_{i=1}^m \lambda_i (\phi_i*\widetilde\phi_i), \qquad \lambda_i\ge0. \] Such a $w$ is entrywise nonnegative, finitely supported, and positive-definite because \[ \widehat w(\theta)=\sum_{i=1}^m \lambda_i\,|\widehat{\phi_i}(\theta)|^2\ge 0. \] Therefore $w$ is admissible, so Theorem Theorem (Universal positive-definite-kernel barrier) applies directly. ∎
Corollary (The barrier is genuinely of order $N^{3/4}$ on near-extremal families): For the infinite family from Round 4 with \[ T=|A|+|B|=(\sqrt2+o(1))\sqrt N, \] the optimum of the whole positive-definite-weight method satisfies \[ \mathcal B_{\mathrm{pd}}(N,T) \ge N+T-1+2\sqrt{T(N-1)} = N+2^{5/4}N^{3/4}+o(N^{3/4}). \] In particular, no proof inside this framework can yield a uniform $N+O(1)$ bound for $S$, and hence none can prove the original $O(1)$-gap conjecture.
Proof: From Theorem Theorem (Universal positive-definite-kernel barrier), \[ \mathcal B_{\mathrm{pd}}(N,T)=\min_{L\ge 1}\left(N+T+L-1+\frac{T(N-1)}{L}\right). \] By AM--GM, \[ L+\frac{T(N-1)}{L}\ge 2\sqrt{T(N-1)}. \] Hence \[ \mathcal B_{\mathrm{pd}}(N,T)\ge N+T-1+2\sqrt{T(N-1)}. \] Substituting $T=(\sqrt2+o(1))\sqrt N$ gives \[ 2\sqrt{T(N-1)}=(2^{5/4}+o(1))N^{3/4}, \] and the result follows. ∎

6) ADVERSARIAL VERIFICATION

(i) Did Round 8 really strengthen Round 7?
Yes. Round 7 treated finite positive combinations of pre-chosen autocorrelation kernels generated by nonnegative functions. Round 8 removes both restrictions: the weight may now be any finitely supported nonnegative positive-definite sequence, equivalently any nonnegative trigonometric polynomial in the frequency side. Theorem Theorem (Universal positive-definite-kernel barrier) therefore subsumes Round 7.
(ii) Hidden hypothesis in Fejér--Riesz.
Theorem Theorem (Fejér--Riesz) applies because $\widehat w$ is a one-variable nonnegative trigonometric polynomial of degree at most $L-1$. This exactly matches the admissible-weight setup.
(iii) Why is $W=\sum_n w(n)$ nonzero?
If $w\not\equiv 0$ and $w(n)\ge0$ for all $n$, then $W>0$. The zero weight yields no information and is excluded from optimisation.
(iv) Equality case in Lemma Lemma (Fejér inequality in coefficient form).
The equality statement uses only the equality case of Cauchy--Schwarz. Once all coefficients $\psi(0),\dots,\psi(L-1)$ have the same phase and modulus, the resulting autocorrelation is exactly a positive multiple of $F_L$. There is no additional hidden symmetry assumption.
(v) Does this settle the first question?
No. It settles only a methodological question: every proof that uses only the pointwise bound and a single nonnegative positive-definite weight (equivalently any finite positive combination of such weights) is now exhausted, and its optimum is exactly known. The first question itself remains open.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED).

Round 8 proves a genuinely stronger barrier theorem than Round 7. The whole nonnegative positive-definite-kernel method is now optimised exactly: \[ \mathcal B_{\mathrm{pd}}(N,T)=\min_{L\ge 1}(N+L-1)\left(1+\frac{T}{L}\right), \] and the unique optimisers are the discrete Fejér kernels. Thus the interval/Fejér kernel was not merely a convenient choice in earlier rounds --- it is the best possible kernel in the entire positive-definite-weight framework. This rules out an even broader proof strategy for the first question.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 93%

9) REFERENCES

  • Round 7 write-up supplied in this task.
  • M. A. Dritschel and J. Rovnyak, The Operator Fejér--Riesz Theorem, for the classical one-variable Fejér--Riesz factorisation statement used here.
  • T. F. Bloom, Erdős Problem #43, problem page and discussion thread.