The Generic Failure

The Littlewood conjecture (1930) asks whether for every pair of real numbers α, β, the product n·‖nα‖·‖nβ‖ can be made arbitrarily small. Here ‖·‖ denotes distance to the nearest integer. The conjecture remains open — one of the oldest in Diophantine approximation. The uniform variant, proposed by Bandi, Fregoli, and Kleinbock, strengthens the question: does the infimum over n ≤ N of the product decrease as a power of 1/N?

The uniform conjecture is false.

Schleischitz (arXiv:2603.12611, March 2026) constructs counterexamples — pairs (α, β) where the infimum does not decrease at the predicted rate. The construction is not delicate. The set of counterexamples is comeagre in ℝ² — meaning it contains a countable intersection of dense open sets. In the topological sense, almost every pair of real numbers violates the conjecture.

The counterexamples likely have positive Hausdorff dimension as well, meaning they are not merely topologically generic but metrically substantial. The conjecture fails not at pathological points but at typical ones.

The structural lesson is about the gap between a conjecture being natural and being true. The uniform Littlewood conjecture is a natural strengthening of a natural conjecture — it asks the obvious quantitative follow-up to a qualitative question. And the qualitative version may still be correct. But the quantitative refinement fails generically: the rate at which approximation improves is slower than the conjecture demands, and this slowness is the rule, not the exception. Strengthening a true statement by a natural quantitative bound can produce a false statement — and the falsity can be maximal.