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On Zaremba’s conjecture. (English) Zbl 1370.11083
S. K. Zaremba [in: Appl. Number Theory numer. Analysis, Proc. Sympos. Univ. Montreal 1971, 39–119 (1972; Zbl 0246.65009)] conjectured that any natural number should occur as a denominator in a convergent of the simple continued fraction of a real number with partial quotients uniformly bounded by some $$A$$. He further speculated that $$A = 5$$ should suffice. In a stronger conjecture, D. Hensley [J. Number Theory 58, No. 1, 9–45 (1996; Zbl 0858.11039)] speculated (wrongly) that if $${\mathcal A}$$ is some finite set of natural numbers for which the Hausdorff dimension of the set of real numbers with all partial quotients in $${\mathcal A}$$ exceeds $$1/2$$, then every sufficiently large natural number should occur as a denominator of a convergent of an element in this set. Hensley’s conjecture is false, as one can easily construct a set $${\mathcal A}$$, for which the denominators of convergents must satisfy certain congruence conditions, ensuring that arithmetic progressions of natural numbers fail to occur.
In the present paper, the authors make spectacular progress on these two conjectures. In the case of Zaremba’s conjecture, it is shown that almost all natural numbers (with respect to density) occur as denominators of convergents to real numbers with partial quotients bounded above by $$50$$. In the case of Hensley’s conjecture, this is modified to take congruence obstructions into account, and the modified conjecture is shown to hold true for almost all integers, albeit for a larger dimensional bound than $$1/2$$. In addition, the authors improve upon previous bounds on the number of natural numbers occurring in the sequence of denominators of convergents of elements in a set of real numbers with partial quotients from a fixed set $${\mathcal A}$$, provided this set has Hausdorff dimension at least $$1/2$$.
The proof uses the circle method by first restating the problem in terms of a semigroup of matrices in $$\text{GL}_2({\mathbb Z})$$ generating the convergents in one entry, and then considering a bilinear form, picking out the denominator. Subsequently, an exponential sum over these bilinear forms is set up and analysed in terms of major and minor arcs. The proof is very technical, but also very well presented.

MSC:
 11J70 Continued fractions and generalizations 11A55 Continued fractions 11B05 Density, gaps, topology
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