The exact logical consequence of the EKL Littlewood exceptional-set theorem

Definitions and mathematical statement

This record is a truncated historical status paragraph, not a self-contained Conjecture 28. It begins mid-sentence, cites [47] concerning Littlewood exceptions, mentions dynamics on $\operatorname{SL}_3({\mathbb{R}})/\operatorname{SL}_3({\mathbb{Z}})$, and ends "It was observed some time ago that" without its continuation. The missing conjecture and conclusion are not reconstructed.

For the background actually discussed, let $\|x\|=\min_{m\in{\mathbb{Z}}}|x-m|$. The classical Littlewood assertion is \[ \forall\alpha,\beta\in{\mathbb{R}},\qquad \liminf_{n\to\infty}n\|n\alpha\|\|n\beta\|=0. \] Let $E$ be its exceptional set, defined by a strictly positive liminf. The aid attributes to the published Einsiedler--Katok--Lindenstrauss theorem a decomposition \[ E=\bigcup_{j\geq1}K_j, \qquad K_j\text{ compact},\quad\overline{\dim}_{\mathrm B}K_j=0, \] in the corresponding bounded-fundamental-domain/exhaustion convention. For bounded sets, upper box dimension is \[ \overline{\dim}_{\mathrm B}K =\limsup_{r\downarrow0}\frac{\log N_r(K)}{\log(1/r)}, \] where $N_r$ is a minimal finite covering number at scale $r$. The reported dimension-theoretic consequence is \[ \dim_{\mathrm H}E=0, \] using $\dim_{\mathrm H}K_j\leq\overline{\dim}_{\mathrm B}K_j$ and countable stability of Hausdorff dimension. A box dimension for an unbounded union would require a separate convention; the local bounded-domain statement avoids that ambiguity.

The aid's examples show exactly what this structural hypothesis does not force. The compact set \[ K=\left\{\sum_{n\geq1}\varepsilon_n2^{-n^2}:\varepsilon_n\in\{0,1\}\right\} \] is nonempty and uncountable with upper box dimension zero. Thus even one compact zero-box-dimensional set need not be empty or countable. On the other hand, ${\mathbb{Q}}\cap[0,1]$ is a countable union of compact singletons but is dense in $[0,1]$, so its upper box dimension is one. Upper box dimension does not have the same countable-union stability as Hausdorff dimension.

These are abstract logical examples, not constructed Littlewood exceptions. The decomposition alone proves neither $E=\varnothing$, countability of $E$, nor upper box dimension zero of the entire union. The aid describes the classical Littlewood conjecture as still open; its dimension theorem and exhaustion consequence are not a solution of that conjecture or of the missing original Conjecture 28.

Short English statement

The supplied background theorem makes the Littlewood exceptional set Hausdorff dimension zero, but does not prove that it is empty or countable. The two explicit sets demonstrate the limits of the decomposition, not actual Littlewood counterexamples. The assigned conjecture fragment remains incomplete.

Sources

• Local UnsolvedMath registry, permanent ID $20003305$, exact source code catalogue entry; canonical aim-other-notes.json, notes[111], AIM domain Other, registry category Miscellaneous, tag conjecture.
• AIM workshop "Emerging applications of measure rigidity," assigned fragment labeled Conjecture 28: https://aimath.org/WWN/measrigid/measrigid.pdf. The record preserves the exact assigned OCR fragment; the aid identifies it as the middle of a historical paragraph. No PDF was fetched.
• Source-local label [47] concerns the Littlewood exceptional-set result; the aid names Einsiedler, Katok, and Lindenstrauss. No title, publication identifier, or full authoritative citation is supplied. The original literature field is empty.
• Research attempt 1: result context_only, problem status at run not_a_problem, rigor label proved, novelty confidence low; classification OPEN-TRIAGE, registry status open, suggested difficulty L2. Created and updated 14 August 2026. These are supplied assessments, not a verdict on an unreproduced conjecture.
• Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. The machine-generated research aid is not peer reviewed and requires independent verification before citation.