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Disproof of the uniform Littlewood conjecture

Johannes Schleischitz

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.

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 . Moreover, we disprove a uniform version of the -adic Littlewood problem, as well as some twisted weaker version of a more general -arithmetic setting, for any proper subset (possible infinite) of primes . The latter contrasts the classical (non-uniform) case where the answer is known to be affirmative when 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.
Paper Structure (14 sections, 18 theorems, 109 equations)

This paper contains 14 sections, 18 theorems, 109 equations.

Table of Contents

  1. On the uniform Littlewood conjecture
  2. A twisted problem
  3. $S$-arithmetic ULC
  4. Some related observations on sumsets
  5. Proof of Theorem \ref{['T01']}: The main lemma
  6. Proof of Lemma \ref{['lemur']}
  7. Deduction of Theorem \ref{['T01']} from Lemma \ref{['lemur']}
  8. Proof of Theorem \ref{['2t']}
  9. Preparatory results
  10. Proof of Theorem \ref{['2t']}
  11. Proof of Theorem \ref{['t2']}
  12. Open problems
  13. Some open problems
  14. On Conjecture \ref{['CCC']}

Key Result

Theorem 1.1

The ULC is false, we have More precisely, the set is a dense $G_{\delta}$ set. Consequently $\Theta$ has full packing dimension and $\Theta+\Theta=\mathbb{R}^2$.

Theorems & Definitions (36)

  • Conjecture 1: ULC
  • Theorem 1.1
  • Remark 1
  • Proposition 1: Erdős
  • Lemma 1.2: icharxiv
  • proof
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • ...and 26 more