On Simultaneous Rational Approximations to a Real Number, Its Square, and Its Cube
نویسنده
چکیده
We show that, for any transcendental real number ξ, the uniform exponent of simultaneous approximation of the triple (ξ, ξ, ξ) by rational numbers with the same denominator is at most (1 + 2γ − √ 1 + 4γ)/2 ∼= 0.4245 where γ = (1 + √ 5)/2 stands for the golden ratio. As a consequence, we get a lower bound on the exponent of approximation of such a number ξ by algebraic integers of degree at most 4.
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