Exploring the Number Jungle: A Journey into Diophantine Analysis

On a quantitative refinement of the Lagrange spectrum

“…Here we quickly review the classical theory of continued fractions (see, for example, [1], Modules 4 and 5). For a real number α, we denote its (simple) continued fraction expansion by α = a 0 + 1…”

“…As we remarked in Section 3, α m ∼ α m , and thus α m ∼ −α m . By a well known identity involving purely periodic continued fractions (see, for instance, [1,Lemma 8.7]), we have In view of (6.4) and (6.3), we now see…”

Some Exact Algebraic Expressions for the Tails of Tasoev Continued Fractions

“…More precisely, D n := q n α − p n = (−1) n α n+1 q n + q n−1 ∀n ≥ 0 (see, for instance, [1,Lemma 5.4]). …”

On diophantine approximation along algebraic curves

“…. , a n ] denote the nth convergent associated with η, then the following is a well-known inequality from the theory of continued fractions (see [2] or [5]):…”

“…A well-known identity from the theory of continued fractions (see [2] or [5]) allows us to rewrite the left-hand side of the previous inequality as…”