Computation of the Infimum in the Littlewood Conjecture

Abstract

The famous Littlewood Conjecture states that for any two real numbers , the value is equal to zero. In this article, we provide an algorithm which for given ε > 0 checks if the value is less than ε. In particular with its help, we show that m_LC < 1/19. We also provide a similar algorithm for p-adic counterpart of the Littlewood Conjecture and show that an analog of m_LC in 2-adic case is at most 1/9.

2000 AMS Subject Classification::

• 11J13

Keywords::

• Littlewood conjecture
• Diophantine approximation
• Farey fractions