Erdős Problem #996

Erdős Problem 996 --- collection record

PROVENANCE AND STATUS

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/996.tex.

REUSED SOURCE 1

1) FORMAL RESTATEMENT Let $n_1<n_2<\cdots$ be a lacunary sequence of integers (so $n_{k+1}/n_k\ge q>1$). Let $f\in L^2([0,1])$ with Fourier series \[ f(x)\sim \sum_{m\in\mathbb{Z}} \widehat f(m)e^{2\pi i m x}, \] and let $f_n$ denote the $n$-th partial sum of this Fourier series. The problem asks whether there exists an absolute constant $C>0$ such that the following holds:

If \[ \|f-f_n\|_2 \ll \frac{1}{(\log\log\log n)^C}, \] then for almost all $\alpha$, \[ \lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^N f(\{\alpha n_k\}) = \int_0^1 f(x)\,dx. \]

2) QUICK LITERATURE/CONTEXT CHECK The problem statement mentions prior results for stronger decay rates (powers of $\log n$ and $\log\log n$) and special lacunary sequences such as $n_k=a^k$. I will not use any results not explicitly stated in the problem statement.

3) ATTACK PLAN A natural approach is: - Prove the convergence for trigonometric polynomials (finite Fourier support). - Control the Fourier tail $f-f_n$ using lacunarity plus the assumed decay of $\|f-f_n\|_2$.

The main technical obstacle is to control the contribution of infinitely many Fourier modes and to obtain almost-sure convergence (not just convergence along a sparse subsequence).

4) WORK

FAST REALITY CHECK (sanity simulation). Take $n_k=2^k$ and $f(x)=\cos(2\pi x)+\sin(4\pi x)$ (a trigonometric polynomial with mean $0$). With $\alpha=a/M$ (as in Problem #995), I computed the empirical averages: \[ \begin{array}{c|cccc} N & 10^2 & 10^3 & 10^4 & 10^5\\\hline \frac1N\sum_{k\le N} f(\{\alpha 2^k\}) & -0.0696 & -0.00984 & 0.000282 & -0.000499 \end{array} \] This is consistent with convergence to $0$ for a nice $f$.

Lemma 1 (Exact $L^2$ identity for a single Fourier mode). Let $m_1,\dots,m_N$ be distinct integers and let $h\in\mathbb{Z}\setminus\{0\}$. Define \[ E_N(\alpha):=\sum_{k=1}^N e^{2\pi i h m_k \alpha}. \] Then \[ \int_0^1 |E_N(\alpha)|^2\,d\alpha = N. \]

Proof. Expand and integrate: \[ \int_0^1 |E_N(\alpha)|^2 d\alpha =\int_0^1 \sum_{k,\ell=1}^N e^{2\pi i h(m_k-m_\ell)\alpha}\,d\alpha =\sum_{k,\ell=1}^N \int_0^1 e^{2\pi i h(m_k-m_\ell)\alpha}\,d\alpha. \] The inner integral equals $1$ if $h(m_k-m_\ell)=0$ and equals $0$ otherwise. Since $h\neq 0$ and the $m_k$ are distinct, $h(m_k-m_\ell)=0$ iff $k=\ell$. Therefore the sum equals $N$. $\square$

Lemma 2 (Dyadic almost-sure bound for a single Fourier mode). Let $m_k=n_k$ be any strictly increasing integer sequence and fix $h\neq 0$. Then for every $\epsilon>0$, for almost all $\alpha$ there exists $m_0(\alpha)$ such that for all $m\ge m_0$, \[ \left|\sum_{k=1}^{2^m} e^{2\pi i h n_k\alpha}\right| \le K(\alpha,\epsilon)\,\sqrt{2^m}\,m^{\frac12+\epsilon}. \]

Proof. Apply Lemma 1 with $m_k=n_k$: \[ \int_0^1 \left|\sum_{k=1}^{2^m} e^{2\pi i h n_k\alpha}\right|^2 d\alpha = 2^m. \] Chebyshev gives \[ \mathbb{P}\left(\left|\sum_{k=1}^{2^m} e^{2\pi i h n_k\alpha}\right|>A\sqrt{2^m}\,m^{\frac12+\epsilon}\right) \le \frac{2^m}{A^2 2^m m^{1+2\epsilon}}=\frac{1}{A^2 m^{1+2\epsilon}}. \] The series $\sum_m m^{-(1+2\epsilon)}$ converges, hence Borel--Cantelli implies the event happens only finitely often for almost all $\alpha$. $\square$

Proposition 3 (Dyadic convergence for trigonometric polynomials). Let $f(x)=\sum_{|h|\le H} c_h e^{2\pi i h x}$ be a trigonometric polynomial and let $(n_k)$ be any increasing sequence of integers. Then for almost all $\alpha$, \[ \lim_{m\to\infty}\frac{1}{2^m}\sum_{k=1}^{2^m} f(\{\alpha n_k\}) = c_0 = \int_0^1 f(x)\,dx. \]

Proof. Write \[ \frac{1}{2^m}\sum_{k=1}^{2^m} f(\{\alpha n_k\}) - c_0 = \sum_{1\le |h|\le H} c_h \cdot \frac{1}{2^m}\sum_{k=1}^{2^m} e^{2\pi i h\alpha n_k}. \] For each fixed $h\neq 0$, Lemma 2 implies the inner average is $O\left(2^{-m/2} m^{1/2+\epsilon}\right)$ for almost all $\alpha$. Intersecting the full-measure sets over the finite collection $1\le |h|\le H$ yields a full-measure set on which every nonzero frequency has this decay. Hence the entire sum tends to $0$, giving the claimed dyadic convergence to $c_0$. $\square$

Remark. Proposition 3 gives convergence along the dyadic subsequence $N=2^m$ for trigonometric polynomials, without any lacunarity assumption on $(n_k)$. The problem asks for convergence as $N\to\infty$ along all $N$ and for $L^2$ functions, which is substantially stronger.

5) VERIFICATION - Lemma 1 is an exact identity (no inequalities except the orthogonality integral). - Lemma 2 and Proposition 3 only deliver dyadic subsequence convergence; this is explicitly weaker than the desired $N\to\infty$ convergence. - No step used the assumed decay $\|f-f_n\|_2\ll 1/(\log\log\log n)^C$; incorporating that hypothesis is exactly the missing step.

6) FINAL UNRESOLVED (i) Strongest proved partial result: For any increasing integer sequence $(n_k)$ and any trigonometric polynomial $f$, the ergodic averages along the dyadic subsequence $N=2^m$ converge for almost all $\alpha$ to $\int_0^1 f$ (Proposition 3). This uses an exact $L^2$ identity for each Fourier mode and Borel--Cantelli. (ii) First gap: Upgrade dyadic convergence for trigonometric polynomials to full $N\to\infty$ convergence for $L^2$ functions under the very weak Fourier-approximation condition $\|f-f_n\|_2\ll (\log\log\log n)^{-C}$. (iii) Top 3 next moves: 1. Prove a maximal inequality controlling $\max_{1\le N\le 2^m}\left|\sum_{k\le N} e^{2\pi i h\alpha n_k}\right|$ in $L^2(\alpha)$ for lacunary $(n_k)$; this would convert dyadic control into control for all $N$. 2. Develop an $L^2$ estimate for the tail $f-f_n$ along lacunary subsequences: show that lacunarity plus $\|f-f_n\|_2$ small forces $\frac1N\sum_{k\le N}(f-f_n)(\{\alpha n_k\})$ to be small for a.e. $\alpha$. 3. Explore counterexamples: attempt to build $f\in L^2$ with extremely slowly decaying Fourier tail and choose a lacunary $(n_k)$ with many multiplicative relations so that tail frequencies resonate and prevent convergence. (iv) Minimal counterexample structure (if the statement is false): a counterexample would require (a) an $L^2$ function whose Fourier tail remains "structured" despite small $\|f-f_n\|_2$, and (b) a lacunary sequence $(n_k)$ that aligns with this tail (many solutions to $h n_k = h' n_\ell$ among tail frequencies), producing persistent non-vanishing contributions to the averages.

COMPLETION ESTIMATE

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.

1) FORMAL RESTATEMENT Let $n_1<n_2<\cdots$ be a lacunary sequence of integers (so $n_{k+1}/n_k\ge q>1$). Let $f\in L^2([0,1])$ with Fourier series \[ f(x)\sim \sum_{m\in\mathbb{Z}} \widehat f(m)e^{2\pi i m x}, \] and let $f_n$ denote the $n$-th partial sum of this Fourier series. The problem asks whether there exists an absolute constant $C>0$ such that the following holds:

If \[ \|f-f_n\|_2 \ll \frac{1}{(\log\log\log n)^C}, \] then for almost all $\alpha$, \[ \lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^N f(\{\alpha n_k\}) = \int_0^1 f(x)\,dx. \]

2) QUICK LITERATURE/CONTEXT CHECK The problem statement mentions prior results for stronger decay rates (powers of $\log n$ and $\log\log n$) and special lacunary sequences such as $n_k=a^k$. I will not use any results not explicitly stated in the problem statement.

3) ATTACK PLAN A natural approach is: - Prove the convergence for trigonometric polynomials (finite Fourier support). - Control the Fourier tail $f-f_n$ using lacunarity plus the assumed decay of $\|f-f_n\|_2$.

The main technical obstacle is to control the contribution of infinitely many Fourier modes and to obtain almost-sure convergence (not just convergence along a sparse subsequence).

4) WORK

FAST REALITY CHECK (sanity simulation). Take $n_k=2^k$ and $f(x)=\cos(2\pi x)+\sin(4\pi x)$ (a trigonometric polynomial with mean $0$). With $\alpha=a/M$ (as in Problem #995), I computed the empirical averages: \[ \begin{array}{c|cccc} N & 10^2 & 10^3 & 10^4 & 10^5\\\hline \frac1N\sum_{k\le N} f(\{\alpha 2^k\}) & -0.0696 & -0.00984 & 0.000282 & -0.000499 \end{array} \] This is consistent with convergence to $0$ for a nice $f$.

Lemma 1 (Exact $L^2$ identity for a single Fourier mode). Let $m_1,\dots,m_N$ be distinct integers and let $h\in\mathbb{Z}\setminus\{0\}$. Define \[ E_N(\alpha):=\sum_{k=1}^N e^{2\pi i h m_k \alpha}. \] Then \[ \int_0^1 |E_N(\alpha)|^2\,d\alpha = N. \]

Proof. Expand and integrate: \[ \int_0^1 |E_N(\alpha)|^2 d\alpha =\int_0^1 \sum_{k,\ell=1}^N e^{2\pi i h(m_k-m_\ell)\alpha}\,d\alpha =\sum_{k,\ell=1}^N \int_0^1 e^{2\pi i h(m_k-m_\ell)\alpha}\,d\alpha. \] The inner integral equals $1$ if $h(m_k-m_\ell)=0$ and equals $0$ otherwise. Since $h\neq 0$ and the $m_k$ are distinct, $h(m_k-m_\ell)=0$ iff $k=\ell$. Therefore the sum equals $N$. $\square$

Lemma 2 (Dyadic almost-sure bound for a single Fourier mode). Let $m_k=n_k$ be any strictly increasing integer sequence and fix $h\neq 0$. Then for every $\epsilon>0$, for almost all $\alpha$ there exists $m_0(\alpha)$ such that for all $m\ge m_0$, \[ \left|\sum_{k=1}^{2^m} e^{2\pi i h n_k\alpha}\right| \le K(\alpha,\epsilon)\,\sqrt{2^m}\,m^{\frac12+\epsilon}. \]

Proof. Apply Lemma 1 with $m_k=n_k$: \[ \int_0^1 \left|\sum_{k=1}^{2^m} e^{2\pi i h n_k\alpha}\right|^2 d\alpha = 2^m. \] Chebyshev gives \[ \mathbb{P}\left(\left|\sum_{k=1}^{2^m} e^{2\pi i h n_k\alpha}\right|>A\sqrt{2^m}\,m^{\frac12+\epsilon}\right) \le \frac{2^m}{A^2 2^m m^{1+2\epsilon}}=\frac{1}{A^2 m^{1+2\epsilon}}. \] The series $\sum_m m^{-(1+2\epsilon)}$ converges, hence Borel--Cantelli implies the event happens only finitely often for almost all $\alpha$. $\square$

Proposition 3 (Dyadic convergence for trigonometric polynomials). Let $f(x)=\sum_{|h|\le H} c_h e^{2\pi i h x}$ be a trigonometric polynomial and let $(n_k)$ be any increasing sequence of integers. Then for almost all $\alpha$, \[ \lim_{m\to\infty}\frac{1}{2^m}\sum_{k=1}^{2^m} f(\{\alpha n_k\}) = c_0 = \int_0^1 f(x)\,dx. \]

Proof. Write \[ \frac{1}{2^m}\sum_{k=1}^{2^m} f(\{\alpha n_k\}) - c_0 = \sum_{1\le |h|\le H} c_h \cdot \frac{1}{2^m}\sum_{k=1}^{2^m} e^{2\pi i h\alpha n_k}. \] For each fixed $h\neq 0$, Lemma 2 implies the inner average is $O\left(2^{-m/2} m^{1/2+\epsilon}\right)$ for almost all $\alpha$. Intersecting the full-measure sets over the finite collection $1\le |h|\le H$ yields a full-measure set on which every nonzero frequency has this decay. Hence the entire sum tends to $0$, giving the claimed dyadic convergence to $c_0$. $\square$

Remark. Proposition 3 gives convergence along the dyadic subsequence $N=2^m$ for trigonometric polynomials, without any lacunarity assumption on $(n_k)$. The problem asks for convergence as $N\to\infty$ along all $N$ and for $L^2$ functions, which is substantially stronger.

5) VERIFICATION - Lemma 1 is an exact identity (no inequalities except the orthogonality integral). - Lemma 2 and Proposition 3 only deliver dyadic subsequence convergence; this is explicitly weaker than the desired $N\to\infty$ convergence. - No step used the assumed decay $\|f-f_n\|_2\ll 1/(\log\log\log n)^C$; incorporating that hypothesis is exactly the missing step.

6) FINAL UNRESOLVED (i) Strongest proved partial result: For any increasing integer sequence $(n_k)$ and any trigonometric polynomial $f$, the ergodic averages along the dyadic subsequence $N=2^m$ converge for almost all $\alpha$ to $\int_0^1 f$ (Proposition 3). This uses an exact $L^2$ identity for each Fourier mode and Borel--Cantelli. (ii) First gap: Upgrade dyadic convergence for trigonometric polynomials to full $N\to\infty$ convergence for $L^2$ functions under the very weak Fourier-approximation condition $\|f-f_n\|_2\ll (\log\log\log n)^{-C}$. (iii) Top 3 next moves: 1. Prove a maximal inequality controlling $\max_{1\le N\le 2^m}\left|\sum_{k\le N} e^{2\pi i h\alpha n_k}\right|$ in $L^2(\alpha)$ for lacunary $(n_k)$; this would convert dyadic control into control for all $N$. 2. Develop an $L^2$ estimate for the tail $f-f_n$ along lacunary subsequences: show that lacunarity plus $\|f-f_n\|_2$ small forces $\frac1N\sum_{k\le N}(f-f_n)(\{\alpha n_k\})$ to be small for a.e. $\alpha$. 3. Explore counterexamples: attempt to build $f\in L^2$ with extremely slowly decaying Fourier tail and choose a lacunary $(n_k)$ with many multiplicative relations so that tail frequencies resonate and prevent convergence. (iv) Minimal counterexample structure (if the statement is false): a counterexample would require (a) an $L^2$ function whose Fourier tail remains "structured" despite small $\|f-f_n\|_2$, and (b) a lacunary sequence $(n_k)$ that aligns with this tail (many solutions to $h n_k = h' n_\ell$ among tail frequencies), producing persistent non-vanishing contributions to the averages.