Disproof of the uniform Littlewood conjecture
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912964562386944 |
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| author | Schleischitz, Johannes |
| author_facet | Schleischitz, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12611 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Disproof of the uniform Littlewood conjecture Schleischitz, Johannes Number Theory 11J13 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. |
| title | Disproof of the uniform Littlewood conjecture |
| topic | Number Theory 11J13 |
| url | https://arxiv.org/abs/2603.12611 |