Almost-everywhere Fourier convergence in L log L

Source and verification

This article explains family 075 of the October 6, 2026 OpenAI collection. The research conclusions are attributed to its manuscripts. The source review covers statements, introductions, and the proof mechanisms described below; it is not an independent verification of the complete proofs.

Overview

Family 075 claims almost-everywhere convergence of ordinary Fourier partial sums for every function in $L\log L$. The endpoint difficulty is to control all frequencies while paying only one logarithm of the input’s size.

🏷️ Preliminaries: Fourier sums and an Orlicz space

Give $\mathbb{T}=\mathbb{R}/(2\pi \mathbb{Z})$ normalized measure $d\mu =dx/(2\pi )$. For an integrable function,

$$\begin{array}{c}\hat{f}(k)={\int }_{\mathbb{T}}f(x)e^{-ikx}d\mu (x),\\ S_{N}f(x)=\sum _{k=-N}^N\hat{f}(k)e^{ikx}.\end{array}$$

The Orlicz class $L\log L$ consists of measurable functions satisfying

$${\int }_{\mathbb{T}}|f|\log (2+|f|)d\mu <\infty .$$

It lies between $L^1$ and the $L^p$ spaces with $p>1$. The extra logarithm penalizes high peaks more than ordinary integrability does, but less than any fixed positive power. Ordinary partial sums must be distinguished from their Cesàro averages: averaging changes the kernel and can improve convergence substantially.

The Carleson maximal operator in this setting is $Cf={\sup }_{N\ge 0}|S_{N}f|$. Controlling each $S_N$ separately does not control this pointwise supremum. The maximizing integer may depend on the spatial point.

🏷️ The manuscript’s endpoint claim

Family 075, as stated in the manuscript

For every complex-valued $f\in L\log L(\mathbb{T})$, the symmetric sums $S_{N}f(x)$ converge to $f(x)$ for almost every $x$, along the entire sequence of integers $N\to \infty$. (OpenAI, 2026)

The exceptional set may depend on $f$. No assertion of convergence for every integrable function is made. The familiar $L^2$ convergence theorem is a useful starting point, but simply letting $p\downarrow 1$ in an $L^p$ theorem does not establish this endpoint: one must track the constants and sum the amplitude levels correctly.

🏷️ The bounded-density estimate

The source first treats a nonnegative density $\rho$ with total mass one and $\rho \le e^d$. Let $M\rho$ be the uncentered interval maximal function. For large $d$, set $h=\log \log d$. Its decisive good-set estimate is

$$\begin{array}{c}{\int }_{\{M\rho \le e^{4h}\}}C\rho d\mu \le Cd,\\ \mu \{M\rho >e^{4h}\}\le C{e}^{-4h}.\end{array}$$

The two bounds serve different purposes. The first says that the analytic cost grows linearly in the logarithm of the height. The second isolates a small spatial set on which that bound is unavailable. To sum many amplitude layers, both the costs and the exceptional-set measures must be summable.

For a nonnegative layer $g$ of mass $m>0$ and height at most $A$, normalization gives $\rho =g/m$ and height $A/m$. Thus the good-set integral cost scales as $m\log (A/m)$, up to the regularization needed for small parameters. This is why logarithmic integrability appears naturally. A complete layer argument must also control the dependence on $m$ and choose heights so that the exceptional sets can be summed; the manuscript uses doubly exponential height layers for this purpose.

🏷️ Why frequency selection costs information

The finite model is a binary tree of dyadic intervals. It estimates modulated Haar differences, stopped when the original density becomes too large. A fixed frequency is comparatively manageable; the problem is to take the supremum over all integer modulations without multiplying the bound by their number.

The manuscript groups frequencies whose phases are nearly indistinguishable under a noisy observation of the input point. An entropy-compression lemma supplies a large group with a common region of phase concentration. Additional observations refine this description, and the argument assigns a quantitative cost to the information they reveal.

Two probability laws appear: the input point is sampled in proportion to $\rho$, while the output path is sampled uniformly. The proof must transfer information estimates under the first law into transform estimates under the second. Its records of representative frequencies are updated before the relevant output choice; this timing is needed for the conditional estimates.

Finally, translation and scale averages of the dyadic kernels recover a Hilbert kernel, and the periodic Hilbert transform reconstructs the symmetric Fourier projections. This transfer is an analytic part of the proof, not an identification of the dyadic and Fourier problems by notation.

🏷️ From maximal control to convergence

Here is the standard final mechanism. Suppose an appropriate weak maximal estimate is available and choose trigonometric polynomials $p_j\to f$ in the associated Orlicz norm. For each polynomial, $S_N{p}_j=p_j$ once $N$ is large. Hence the limiting oscillation of $S_{N}f$ is bounded by $2C(f-p_j)$. A weak estimate makes the set with oscillation greater than a fixed positive number shrink to measure zero as $j\to \infty$. Approximation of $f$ itself then identifies the limit.

This density argument is elementary once the maximal theorem has been established. It does not replace the endpoint bounded-density, compression, and layer estimates.

🏷️ Key steps and proof outlines

Delayed signs isolate a predictable martingale estimate

The dyadic model has two random points: an input $Y$ weighted by the density and an output $x$ sampled uniformly. At depth $j$, its coefficient averages a modulated Haar difference of the input over the dyadic parent containing $x$. The output factor uses a later sign, $r_{j+2}(x)$. This delay allows the coefficient and its multiplier to be known before the sign is revealed.

The elementary estimate behind this design is the following. If $r_{i+1}$ are conditionally fair signs, $c_i$ is real and known at time $i$, and $\theta >0$, then

$$Z_n=\exp \left(\theta \sum _{i<n}{c}_i{r}_{i+1}-\frac{{\theta }^2}{2}\sum _{i<n}{c}_i^2\right)$$

is a nonnegative supermartingale. Indeed, conditioning one factor uses $\cosh (\theta c_i)\le \exp ({\theta }^2{c}_i^2/2)$. Stopping at the first crossing of $e^t$ gives, with failure probability at most $e^{-t}$, simultaneously for all prefixes,

$$\sum _{i<n}{c}_i{r}_{i+1}\le \frac{\theta }{2}\sum _{i<n}{c}_i^2+\frac{t}{\theta }.$$

This is the source’s predictable-sign lemma. The entire frequency-selection procedure must respect its measurability assumption. Choosing a coefficient after seeing the sign would invalidate the calculation. (OpenAI, 2026)

Compression supplies common representatives for many frequencies

A union bound over all modulations would introduce a dependence on the number of frequencies, which is unacceptable for the maximal operator. The entropy-compression lemma instead extracts a large subset $S^{\prime }$ of frequencies and a center $s_0$ so that their phase differences concentrate on a common region. Its conclusion is uniform over probability distributions on $S^{\prime }$. This uniformity permits a later stage of the procedure to choose which remaining frequencies matter without rerunning an uncontrolled selection.

The proof builds correlated letter distributions and measures the information gained by revealing block sums. A chain-rule identity expresses the entropy lost by one letter as the difference between two block-sum entropies. A gluing argument shows that the same observation decreases a noisy cost by a corresponding amount. Repeating efficient observations therefore spends a finite entropy budget. When no efficient observation remains, the residual set has the common phase concentration required for compression. The stopping alternative, rather than a direct count of frequencies, is what makes the bound independent of the frequency cutoff.

Thermal kernels make information costs quantitative

The manuscript encodes phase observations with positive kernels of the form

$$\exp \left[-\sum _s{\lambda }_s(1-\cos (2\pi s\theta ))\right],{\lambda }_s\ge 0.$$

Each factor has nonnegative Fourier coefficients: expand the exponentials of $e^{2\pi is\theta }$ and $e^{-2\pi is\theta }$ separately. Absolute convergence permits multiplication. Positivity lets the proof compare kernels as more observations are added. For the resulting exponent $A(\theta )$, normalize to the probability density $K_A=e^{-A}/\int e^{-A}$. A two-copy covariance identity writes the relevant covariance as a sum of squares, giving monotonicity of the Fourier coefficients of $K_A$ under increasing the parameters ${\lambda }_s$. This monotonicity concerns the normalized density; the raw kernel’s zeroth coefficient can decrease.

These kernel inequalities connect phase information with a Fisher-type quadratic quantity. The local budget controls sums of squared changes in conditional mass divided by current mass. That is exactly the form needed to bound the quadratic variation in the predictable-sign argument. Zero-mass terms are defined as zero, and all densities are taken relative to a common dominating measure; these conventions avoid dividing by a vanished conditional probability.

Assembly and transfer back to ordinary Fourier sums

On each density plateau, the proof freezes representative frequencies at predictable snapshots, bounds their transforms using the sign estimate, and charges changes of representative to the information budget. Static plateaus, low-mass pieces, and rare transfers have separate bounds. Their sum gives a bound uniform in both dyadic depth and modulation cutoff. The stopping rule is always defined using the original density, so different frequency representatives are not being compared on incompatible good sets.

To recover the continuous kernel, the source averages translated dyadic Haar profiles over translation and scale. Translation converts the pair of input/output profiles into a convolution kernel; the scale average identifies a multiple of the Hilbert kernel. Modulation and the periodic Hilbert transform then recover the symmetric Fourier projections. Uniformity in the finite cutoffs permits the limiting supremum.

Finally, normalize each amplitude layer by its mass. Its analytic cost is controlled by its mass times a logarithm of normalized height, while its bad-set cost is controlled separately by the maximal theorem. Doubly exponential layers make the latter costs summable. This is the specific point where the endpoint $L\log L$ hypothesis is used; the martingale and compression arguments alone are still bounded-input estimates. The resulting maximal control and polynomial approximation give almost-everywhere convergence by the oscillation argument already described.

See Also

• Family 080: Schrödinger convergence also passes from a maximal estimate and smooth approximation to convergence on one full-measure spatial set.
• Family 082: triangular Hilbert transforms distinguishes uniform truncation bounds, maximal bounds, and variation bounds.

🏷️ Manuscripts and verification scope

• Almost-everywhere Fourier convergence in $L\log L$ — the pinned manuscript; its directory supplies the source and citation metadata.

No family-specific scope page was available at lean/docs/075.md in this snapshot. This records the catalogue lookup, not a claim that no related Lean code exists. No Lean build or axiom audit was performed.

OpenAI collection gives the reading sequence and source policy; coverage register records which families have articles and which remain to be written.

🐻  OpenAI 2026. Almost-everywhere Fourier convergence in L log L.