SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES Academic Article uri icon

abstract

  • In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence ${mathcal{D}}$, we obtain a sharp criterion such that for almost every $unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{mathcal{D}}}|nunicode[STIX]{x1D6FC}-p|leq unicode[STIX]{x1D713}(n)end{eqnarray}$$ has infinitely many coprime solutions $(n,p)in mathbb{N} imes mathbb{Z}$ for a certain one-parameter family of $unicode[STIX]{x1D713}$. Also, under a minor condition on pseudo-absolute-value sequences ${mathcal{D}}_{1},{mathcal{D}}_{2},ldots ,{mathcal{D}}_{k}$, we obtain a sharp criterion on a general sequence $unicode[STIX]{x1D713}(n)$ such that for almost every $unicode[STIX]{x1D6FC}$ the inequality $$\begin{eqnarray}|n|_{{mathcal{D}}_{1}}|n|_{{mathcal{D}}_{2}}cdots |n|_{{mathcal{D}}_{k}}|nunicode[STIX]{x1D6FC}-p|leq unicode[STIX]{x1D713}(n)end{eqnarray}$$ has infinitely many coprime solutions $(n,p)in mathbb{N} imes mathbb{Z}$.

published proceedings

  • JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY

altmetric score

  • 1.25

author list (cited authors)

  • Liu, W.

citation count

  • 0

complete list of authors

  • Liu, Wencai

publication date

  • August 2019