ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

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On Khintchine exponents and Lyapunov exponents of continued fractions (2008, background): “A more explicit result due to Borel-Bernstein (see [2, 5, 6]) is the 0-1 law which hints that for almost all x ∈ [0, 1], an(x) > ϕ(n) holds for infinitely many n’s or finitely many n’s according to ∑ n≥1 1 ϕ(n) diverges or converges.”
Hausdorff dimension of certain sets arising in continued fraction expansions (2008, methodology): “These include the work of Bumby [6,7], Hensley [14,15], Pollicott and Weiss [28], Jenkinson and Pollicott [20], Jenkinson [19], Mauldin and Urbański [25], etc.”
Zaremba’s conjecture (2011, background): “21), whereas the alphabet A = {1, 2, 3, 4, 5} is known [Jen04] to have dimension δA > 5/6.”