Law-of-Iterated-Logarithm Bounds for Dyadic Martingales

Definitions and mathematical statement

Let $I_k(x)$ denote the nested dyadic intervals containing $x$, with $|I_k(x)|=2^{-k}$. For example, the standard half-open dyadic grid gives \[ I_k(x)=\left[2^{-k}\lfloor 2^kx\rfloor,\,2^{-k}(\lfloor 2^kx\rfloor+1)\right). \] This specifies a conventional interpretation of the interval notation; the supplied record does not specify a grid or endpoint convention.

For a function $f$ integrable on an interval $I$, its interval mean is \[ f_I=\frac{1}{|I|}\int_I f(t)\,{\,\mathrm{d}} t. \] For real-valued $f$ with these means defined, set \[ d_k(x)=f_{I_{k+1}(x)}-f_{I_k(x)},\qquad Q_n(x)=\sum_{k=n}^{\infty}d_k(x)^2\in[0,+\infty]. \] Real-valuedness makes this series nonnegative. The source does not specify the scalar field or the admissible function class; real-valued local integrability is an interpretation sufficient to define these means, not a confirmed complete hypothesis of the intended problem.

The supplied question asks whether, for every function in the intended but unspecified admissible class, \[ \limsup_{n\to\infty} \frac{\left|f_{I_n(x)}-f(x)\right|} {\sqrt{Q_n(x)\,\left|\log\left|\log Q_n(x)\right|\right|}} <+\infty \qquad\text{for Lebesgue-almost every }x\in{\mathbb{R}}. \] Here $\log$ denotes the natural logarithm. Finiteness is pointwise almost everywhere; the displayed assertion does not require a common bound independent of $x$ or $f$.

Unresolved qualifications. The record supplies neither the function class nor conventions making the quotient meaningful at all relevant indices. In particular, the expression is undefined at $Q_n(x)=0$, $Q_n(x)=1$, or $Q_n(x)=+\infty$, and its denominator vanishes at $Q_n(x)=e^{-1}$ or $Q_n(x)=e$. A condition such as $0<Q_n(x)<e^{-1}$ eventually would make the displayed quotient well-defined, but is not supplied and is not imposed here. The case of eventually vanishing square-function tails requires an explicit convention or a separate formulation. Consequently, this is a statement template preserving the supplied question, pending recovery of its missing hypotheses and boundary conventions.

Short English statement

The question attributed in the supplied curation to Nicolau asks whether the error between a function and its dyadic interval means has a finite almost-everywhere limsup after normalization by its remaining dyadic square function and an iterated logarithm. The function class and conventions for degenerate square-function tails are missing, so the intended assertion requires clarification.

Sources

• Local UnsolvedMath registry, permanent ID 30001617, source problem code catalogue entry, title: Law-of-Iterated-Logarithm Bounds for Dyadic Martingales. The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2010.
• Supplied source citation: Operator Theory and Harmonic Analysis (2011), Oberwolfach Reports, pp. 2813--2875, DOI 10.4171/owr/2010/49. Source URL: https://doi.org/10.4171/owr/2010/49. The background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Attribution: the supplied literature assessment describes this as the dyadic martingale square-function limsup question recorded by Nicolau. The proposer field is empty; the record does not establish that Nicolau proposed it.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2010/49.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports related square-function estimates but no proof of this exact almost-everywhere assertion in the checked literature. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with a recorded manual semantic, typographic, and LaTeX review describing the cleanup as meaning-preserving and self-contained. These are supplied review labels; they do not resolve the missing hypotheses identified here. The report has not been independently consulted for this draft.