A mathematical preprint reports that counterexamples to the uniform Littlewood conjecture can form positive-dimensional families. It studies U, the set of counterexample pairs of real numbers, and proves that the pairs whose first coordinate is badly approximable have Hausdorff dimension at least 3/2. The paper uses Hausdorff dimension to express the size of these constructed sets. That figure is a lower bound, not an exact dimension for the pair set.
This is a proof about mathematical sets, not an empirical study. It analyzes sets of real numbers built from continued fractions, so there are no participants, observations or dataset behind the result. The sets include F_A, defined by bounded continued-fraction partial quotients, and Bad_1, the set of numbers with that bounded-partial-quotient property. The paper asks whether counterexamples occur on sets of positive Hausdorff dimension, whether one can have a badly approximable first coordinate, and how large the qualifying first-coordinate projection can be.
Counting the building blocks
At the core of the proof is Hensley’s counting theorem for a fixed continued-fraction alphabet. For A at least 2, the counting input gives, at dyadic denominators, a number of reduced rationals of order T raised to the power 2 delta_A as T tends to infinity. The parameter delta_A is the Hausdorff dimension associated with F_A. The proof combines this count with a product-set estimate that works for arbitrary composite moduli.
That modular estimate bounds the multipliers that would send every element of a chosen set M outside a fixed interval. The proof also uses Fourier inversion and Vinogradov’s bilinear estimate. It then transfers bounds from rational approximants to real pairs using the fact that the map taking x to its distance from the nearest integer is 1-Lipschitz. That means the quantity changes by no more than the underlying number changes.
Turning constructions into dimension bounds
To turn those estimates into a dimension bound, the proof constructs compact sets of candidate pairs and applies the mass distribution principle. This is the dimension argument behind the paper’s positive-dimensional counterexample families.
For A meeting the theorem’s stated condition, the paper proves that U intersected with F_A cross the real line has Hausdorff dimension at least twice delta_A minus one-half. In plain terms, restricting the first coordinate to F_A still leaves a substantial set of counterexample pairs. The result remains conditional on that stated condition and is a lower bound rather than an exact value.
When the first coordinate is required to lie in Bad_1, the counterexample pairs have Hausdorff dimension at least 3/2. The paper also establishes existence at the level of individual pairs: there are real numbers x and y, with x in Bad_1, for which the uniform Littlewood limsup is positive. Equivalently, U intersected with Bad_1 cross the real line is nonempty. This existence result gives no frequency or measure estimate.
The first coordinate tells a different story
The paper then looks at projections, meaning it keeps only the first coordinate of a qualifying pair. P_A is the set of x in F_A for which some second coordinate forms a counterexample, while P is the analogous set restricted to Bad_1. For every A at least 2 with delta_A greater than 3/4, P_A has Hausdorff dimension delta_A. The full set P has Hausdorff dimension 1.
The full-dimension conclusion is local as well. In every nonempty bounded open interval, the first-coordinate set P intersected with that interval has Hausdorff dimension 1. This is a statement about the first-coordinate projection, not the two-coordinate counterexample set. It says that qualifying first coordinates remain full-dimensional within each bounded interval covered by the theorem.
The result’s boundaries
The distinctions matter when reading the headline result. A dimension of 1 for P, or for its intersection with a bounded interval, should not be mistaken for a dimension of 1 for U as a set of pairs. Likewise, the 3/2 figure for pairs with a badly approximable first coordinate is a lower bound. The work is deterministic mathematics, with no empirical population or causal intervention.
Publication status
The supplied document is arXiv preprint version 1, dated 25 August 2026.
Paper data and sources
Original title: The uniform Littlewood conjecture fails on a set of positive Hausdorff dimension
Authors: Nikita Shulga
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text