On Two Exponents of Approximation Related to a Real Number and Its Square

Content Provider	Semantic Scholar
Author	Roy, Damien
Copyright Year	2007
Abstract	For each real number ξ, let λ̂2(ξ) denote the supremum of all real numbers λ such that, for each sufficiently large X, the inequalities |x0| ≤ X, |x0ξ − x1| ≤ X−λ and |x0ξ − x2| ≤ X−λ admit a solution in integers x0, x1 and x2 not all zero, and let ω̂2(ξ) denote the supremum of all real numbers ω such that, for each sufficiently large X, the dual inequalities |x0 + x1ξ + x2ξ| ≤ X−ω , |x1| ≤ X and |x2| ≤ X admit a solution in integers x0, x1 and x2 not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponentsλ̂2(ξ) where ξ ranges through all real numbers with [Q(ξ) : Q] > 2 form a dense subset of the interval [1/2, ( √ 5 − 1)/2] while, for the same values of ξ, the dual exponents ω̂2(ξ) form a dense subset of [2, ( √ 5 + 3)/2]. Part of the proof rests on a result of V. Jarnı́k showing that λ̂2(ξ) = 1 − ω̂2(ξ)−1 for any real number ξ with [Q(ξ) : Q] > 2.
File Format	PDF HTM / HTML
Alternate Webpage(s)	http://arxiv.org/pdf/math/0409232v1.pdf
Alternate Webpage(s)	http://aix1.uottawa.ca/~droy/papers/roy_CJM_2007.pdf
Alternate Webpage(s)	https://cms.math.ca/cjm/abstract/pdf/149984.pdf
Alternate Webpage(s)	http://arxiv.org/pdf/math/0409232v2.pdf
Language	English
Access Restriction	Open
Subject Keyword	Arabic numeral 0 Dual Rest Status Epilepticus Subgroup
Content Type	Text
Resource Type	Article