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作者:R. Daniel Mauldin , Mariusz Urbanski
来源:[J].tran(IF 1.019), 1999, Vol.351 (12), pp.4995-5025
摘要:... We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmeti...
作者:Oleg Karpenkov , Matty van Son
来源:[J].Journal of Number Theory(IF 0.466), 2020
摘要:... The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers.
作者:R. Daniel Mauldin , Mariusz Urbański
来源:[J].Transactions of the American Mathematical Society(IF 1.019), 1999, Vol.351 (12), pp.4995-5025
摘要:... We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmeti...
作者:Vladimir I. Arnold
来源:[J].Functional Analysis and Other Mathematics, 2009, Vol.2 (2-4), pp.129-138
摘要:Abstract(#br)The article describes the interrelations between the minimal integer number N ( a , b , c ) which belongs to the additive semigroup of integers generated by a , b , c together with all greater integers, on the one hand, and the geometrical theory of continued fractions...
作者:Oleg Karpenkov , Matty van-Son
来源:[J].Journal de théorie des nombres de Bordeaux, 2019, Vol.31 (1), pp.131-144
摘要:... We express the values of binary quadratic forms with positive discriminant in terms of continued fractions associated to broken lines passing through the points where the values are computed.
作者:Patrick Popescu-Pampu
来源:[C].Singularities in Geometry and Topology 20042007
摘要:We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the...
作者:Oleg Karpenkov
来源:[B].Algorithms and Computation in Mathematics;;Springer Textbook2013
摘要:Abstract In the beginning of this book we discussed the geometric interpretation of regular continued fractions in terms of LLS sequences of sails. Is there a natural extension of this interpretation to the case of continued fractions with arbitrary elements? The aim of this cha...
作者:Oleg Karpenkov
来源:[B].Algorithms and Computation in Mathematics2013
摘要:Abstract Continued fractions play an important role in the geometry of numbers. In this chapter we describe a classical geometric interpretation of regular continued fractions in terms of integer lengths of edges and indices of angles for the boundaries of convex hulls of all int...
作者:Oleg Karpenkov
来源:[B].Algorithms and Computation in Mathematics;;Cambridge Studies in Advanced Mathematics;;Series on Knots and Everything2013
摘要:Abstract It turns out that the frequency of a positive integer k in a continued fraction for almost all real numbers is equal to $$\frac{1}{\ln2}\ln \biggl(1+\frac{1}{k(k+2)} \biggr), $$ i.e., for a general real x we have 42 % of 1, 17 % of 2, 9 % of 3, etc. This distribution i...

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