Honam Mathematical Journal (호남수학학술지)

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A FUNCTION CONTAINING ALL LAGRANGE NUMBERS LESS THAN THREE

  • DoYong Kwon (Department of Mathematics, Chonnam National University)
  • Received : 2023.02.08
  • Accepted : 2023.04.24
  • Published : 2023.09.14
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Abstract

Given a real number α, the Lagrange number of α is the supremum of all real numbers L > 0 for which the inequality |α - p/q| < (Lq2)-1 holds for infinitely many rational numbers p/q. All Lagrange numbers less than 3 can be arranged as a set {lp/q : p/q ∈ ℚ ∩ [0, 1]} using the Farey index. The present paper considers a function C(α) devised from Sturmian words. We demonstrate that the function C(α) contains all information on Lagrange numbers less than 3. More precisely, we prove that for any real number α ∈ (0, 1], the value C(α) - C(0) is equal to the sum of all numbers 3 - lp/q where the Farey index p/q is less than α.

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Acknowledgement

The comments by anonymous referees, which made the paper readable, are gratefully acknowledged. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B02010219).

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