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An approximation property of quadratic irrationals

Takao Komatsu (2002)

Bulletin de la Société Mathématique de France

Let α > 1 be irrational. Several authors studied the numbers ℓ m ( α ) = inf { | y | : y ∈ Λ m , y ≠ 0 } , where m is a positive integer and Λ m denotes the set of all real numbers of the form y = ϵ 0 α n + ϵ 1 α n - 1 + ⋯ + ϵ n - 1 α + ϵ n with restricted integer coefficients | ϵ i | ≤ m . The value of ℓ 1 ( α ) was determined for many particular Pisot numbers and ℓ m ( α ) for the golden number. In this paper the value of  ℓ m ( α ) is determined for irrational numbers  α , satisfying α 2 = a α ± 1 with a positive integer a .

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