Disproof of the uniform Littlewood conjecture

Abstract

We show that the uniform Littlewood Conjecture (ULC) recently introduced by Bandi, Fregoli and Kleinbock is false. More precisely the counterexamples form a residual set, the method further suggests positive Hausdorff dimension. For a mildly twisted problem, we indeed separately show that the Hausdorff dimension is at least $1$. Moreover, we disprove a uniform version of the $p$-adic Littlewood problem, as well as some twisted weaker version of a more general $S$-arithmetic setting, for any proper subset (possible infinite) of primes $S$. The latter contrasts the classical (non-uniform) case where the answer is known to be affirmative when $S$ has at least two elements. The disproof of ULC, our main new result, is semi-constructive; the non-constructive part involves effective results on Zaremba's famous conjecture by Bourgain and Kontorovich, as well as estimates for the cardinality of product sets over finite fields.