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Bibliography on simultaneous Diophantine approximation

[1] J. F. Koksma. Diophantische Approximationen. Chelsea, New York, reprint, 19??
[2] A. Broise. Fractions continués multidimensionelles et lois stables. Bulletin de la Société mathématique de France, 124:97-139, 1996.
[3] T. N. Langtry. An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules. Mathematics of computation, 65:1635-1662, 1996.
[4] M. M. Dodson and J. A. G. Vickers, editors. Number theory and dynamical systems, volume 134 of London mathematical society lecture note series. CUP, Cambridge, 1989.
[5] V. I. Bernik and M. M. Dodson. Metric Diophantine approximation on manifolds. CUP, 1999.
[6] F. Schweiger. Ergodic theory of fibred systems and metric number theory. Clarendon Press, Oxford, 1995.
[7] Stefano Marmi. An introduction to small divisors problems. Istituti editoriali e poligrafici internazionali, Pisa, Roma, 2000. ISBN 88-8147-227-9.
[8] Afonso Ferreira, Jérôme Galtier, and Stéphane Perennes. Approximation of a straight line in a bounded lattice. Technical report, INRIA Sophia Antipolis, 2000. ALGOTEL2000.
[9] M. L. Kontsevich and Yu. M. Suhov. Statistics of Klein polyhedra and multidimensional continued fractions. In Pseudoperiodic topology, volume 197 of Am. Math. Soc. Transl., pages 9-27. 1999. Dedicated to V. I. Arnold on his 60th anniversary.
[10] John. D. Hobby. A natural lattice basis problem with applications. ?, ?:?, ? Bell Labs preprint.
[11] N. G. Moshchevitin. Continued fractions, multidimensional Diophantine approximations and applications. J. de Théorie des Nombres de Bordeaux, 11:425-438, 1999. www.emis.de/journals/JNTB.
[12] Edward B. Burger. On real quadratic number fields and simultaneous Diophantine approximation. Monatshefte für Mathematik, 128:201-209, 1999.
[13] Thilo Dienst. On a problem of Schoenberg and Wills in Diophantine approximation. Period. Math. Hung., 36:105-118, 1998.
[14] D. M. Hardcastle and K. Khanin. Almost everywhere strong convergence of multidimensional continued fraction algorithms. Technical Report HPL-BRIMS-00-12, BRIMS, Bristol, UK, 2000.
[15] D. M. Hardcastle and K. Khanin. Continued fractions and the d-dimensional Gauss transform. Technical Report HPL-BRIMS-00-15, BRIMS, Bristol, UK, 2000.
[16] A. D. Bruno. Local methods in nonlinear differential equations: Part I: The local method of nonlinear analysis of differential equations; Part II: The sets of analyticity of a normalizing transformation. Soviet Mathematics. Springer, 1989.
[17] M. R. Herman. Recent results and some open questions on Siegel's linearization theorem of germs of complex analytic diffeomorphisms of n near a fixed point. In M. Mebkhout and R. Sénéor, editors, Proc. VIII Int. Congress Math. Phys., Marseille, July 16-25 1986, pages 138-184. World Scientific, c1987.
[18] K. M. Briggs. On the Furtwängler algorithm for simultaneous rational approximation. Exp. Math. (to be submitted), ?:?, 2001.
[19] P. Arnoux and A. Nogueira. Mesures de gauss pour des algorithmes de fractions continués multidimensionelles. Annales scientifiques de l'école normale supérieure, 26:645-664, 1993. MR 95h:11076.
[20] E. Korkina. The periodicity of multidimensional continued fractions. C. R. Acad. Sci., 319:777-780, 1994. MR 95j:11064.
[21] A. D. Bryuno and V. I. Parusnikov. Klein polyhedra for two cubic Davenport forms. Mathematical notes, 56(3-4):9-27, 1994. Keldysh Institute of the RAS, preprint 48.
[22] A. D. Bryuno and V. I. Parusnikov. Comparison of various generalizations of continued fractions. Mathematical notes, 61:278-286, 1997. Keldysh Institute of the RAS, preprint 52.
[23] Gilles Lachaud. Polyèdre d'Arnol'd et voile d'un cône simplicial: analogues du théorème de Lagrange. C. R. Acad. Sci. Paris, 317:711-716, 1993.
[24] Gilles Lachaud. Klein polygons and geometric diagrams. In Contemporary mathematics, volume 210, pages 365-372. 1998. MR 99a:11086.
[25] Gilles Lachaud. Sails and Klein polyhedra. In Contemporary mathematics, volume 210, pages 373-385. 1998. MR 98k:11094.
[26] E. I. Korkina. Two-dimensional continued fractions. The simplest examples. Proc. Steklov Institute of Mathematics, 209:124-144, 1995. MR 97k:11104.
[27] V. I. Arnold. Higher dimensional continued fractions. Regular and chaotic dynamics, 3:10-17, 1998. MR 2000h:11012; web.uni.udm.ru/rcd/rcd/index.html.
[28] A. D. Bruno. A new generalization of the continued fraction (Russian). Technical Report 82, Keldysh Institute of the RAS, Moscow, 1999.
[29] V. I. Parusnikov. Klein's polyhedra for the third extremal ternary cubic form (Russian). Technical Report 137, Keldysh Institute of the RAS, Moscow, 1995.
[30] V. I. Parusnikov. Klein's polyhedra with big faces (Russian). Technical Report 93, Keldysh Institute of the RAS, Moscow, 1997.
[31] V. I. Parusnikov. Klein's polyhedra for the fifth extremal cubic form (Russian). Technical Report 69, Keldysh Institute of the RAS, Moscow, 1998.
[32] V. I. Parusnikov. Klein's polyhedra for the seventh extremal cubic form (Russian). Technical Report 79, Keldysh Institute of the RAS, Moscow, 1999.
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[34] I. V. L. Clarkson, J. E. Perkins, and I. M. Y. Mareels. On the novel application of number theoretic methods to radar detection. In Proceedings of the international conference on signal processing applications and technology (ICSPAT 93), 1993.
[35] Vaughan Clarkson, Jane Perkins, and Iven Mareels. An algorithm for best approximation of a line by lattice points in three dimensions. Technical report, 1995. 3rd Conference on Computational Algebra and Number Theory (CANT 95), Conference abstracts and other information. Formerly online at wwwcrasys.anu.edu.au/Projects/pulseTrain/Papers/CPM95.ps.gz.
[36] I. V. L. Clarkson, J. E. Perkins, and I. M. Y. Mareels. An algorithm for best approximation of a line by lattice points in three dimensions. Technical report, 1995. presented at 19th Journées Arithmetique, Barcelona.
[37] I. V. L. Clarkson, S. D. Howard, and I. M. Y. Mareels. Estimating the period of a pulse train from a set of sparse, noisy measurements. In Proceedings of the international symposium on signal processing applications and its aplications (ISSPA 96), volume 2, pages 885-888, 1996.
[38] I. V. L. Clarkson, J. E. Perkins, and I. M. Y. Mareels. Number/theoretic [sic] solutions to intercept time problems. IEEE Trans. Information Theory, 42:959-971, 1996.
[39] I. Vaughan L. Clarkson and Iven M. Y. Mareels. Finding best simultaneous diophantine approximations using sequences of minimal sets of lattice points. Technical report, Submitted to Math. Comp., 1996. rejected? Formerly at wwwcrasys.anu.edu.au/Projects/pulseTrain/Papers/CM96.ps.gz.
[40] I. Vaughan L. Clarkson. Approximation of Linear Forms by Lattice Points, with applications to signal processing. PhD thesis, Australian National University, 1997.
[41] C. Rössner and C. P. Schnorr. An optimal, stable continued fraction algorithm for arbitrary dimension. Electronic Colloquium on computational complexity, TR96-020:1-14, 1996.
[42] J. Hastad, B. Just, J. C. Lagarias, and C. P. Schnorr. Algorithms for finding integer relations among real numbers. SIAM J. Comp., 18:859-881, 1989.
[43] B. Just. Generalizing the continued fraction algorithm to arbitrary dimensions. SIAM J. Computing, 21:909-926, 1992.
[44] H. Minkowski. Zur theorie der Kettenbrüche. In David Hilbert, editor, Gesammelte Abhandlungen, volume 1, pages 278-292. 1911. Reprinted Chelsea Pub. Co. 1967.
[45] H. Minkowski. Zur geometrie der Zahlen. In David Hilbert, editor, Gesammelte Abhandlungen, volume 2, pages 43-52. 1911. reprinted Chelsea Pub. Co. 1967.
[46] J. R. Kinney. Note on a singular function of Minkowski. Proc. Am. Math. Soc., 11:788, 1960.
[47] Pelagro Viader and Jaume Paradis. A new light on Minkowski's ?(x) function. Journal of Number Theory, 73:212-227, 1998.
[48] Roland Girgensohn. Constructing singular functions via Farey fractions. Journal of Mathematical Analysis and Applications, 203:127-141, 1996.
[49] Werner Kratz. On optimal constants for best two-dimensional simultaneous diophantine approximations. Monatshefte für Mathematik, 128:99-110, 1999.
[50] W. G. Nowak. A note on simultaneous Diophantine approximation. Manuscripta Math., 36:33-46, 1981. MR 83a:10062.
[51] W. G. Nowak. A remark concerning the s-dimensional simultaneous Diophantine approximation constants. Grazer Math. Ber., 318:105-110, 1992.
[52] Werner Georg Nowak. The critical determinant of the double paraboloid and Diophantine approximation on R3. Technical report, Universität für Bodenkultur Wien, no date.
[53] Werner Georg Nowak. Diophantine approximation on Rs: on a method of Mordell and Armitage. Technical report, Universität für Bodenkultur Wien, 1999.
[54] Werner Georg Nowak. The critical determinant of the double paraboloid and Diophantine approximation on R3 and R4. Mathematica Pannonica, 10:111-122, 1999.
[55] E. M. Bollt and J. D. Meiss. Breakup of invariant tori for the 4-dimensional semistandard map. Physica D, 66:282-297, 1993.
[56] S. Krass. Estimates for n-dimensional diophantine approximation constants for n 4. J. Num. Th., 20:172-176, 1985.
[57] T. W. Cusick and S. Krass. Formulas for some diophantine approximation constants. J. Austral. Math. Soc., A44:311-323, 1988.
[58] S. Krass. The n-dimensional diophantine approximation constants. Bulletin of the Australian Mathematical Society, 32:313-316, 1985.
[59] Victor Shoup. NTL http://www.shoup.net/ntl/. 2000.
[60] W. W. Schmidt. Diophantine Approximation, volume 785 of Lecture Notes in Mathematics. Springer-Verlag, first edition, 1996. Second printing.
[61] Mélanie Crespo. Exposants de Lyapounov pour algorithmes de fractions continues: une extension aux dimensions supérieuses. Master's thesis, Université de Genève, Faculté des Sciences, Section de Mathématiques, 1998.
[62] P. R. Baldwin. A multidimensional continued-fraction and some of its properties. J. Stat. Phys., 66:1463-1505, 1992.
[63] P. R. Baldwin. A convergence exponent for multidimensional continued-fraction algorithms. J. Stat. Phys., 66:1507-1526, 1992.
[64] S. Ito, M. Keane, and M. Ohtsuki. Almost everywhere exponential convergence of the modified Jacobi-Perron algorithm. Ergodic Theory and Dynamical Systems, 13:319-334, 1993.
[65] David J. Grabiner. Farey nets and multidimensional continued fractions. Monatshefte für Mathematik, 114:35-60, 1992.
[66] Max Bauer. Multidimensional continued fractions and the topological entropy of pseudo-anosov maps. Technical Report 95-09, Institut de Recherche Mathématique de Rennes, 1995.
[67] A. Nogueira. The three-dimensional Poincaré continued fraction algorithm. Israel J. Math., 90:373-401, 1995.
[68] V. Baladi and A. Nogueira. Lyapunov exponents for non-classical multidimensional continued fraction algorithms. Nonlinearity, 9:1529-1546, 1996.
[69] Ph. Furtwängler. Über die simultane Approximation von Irrationalzahlen (Erste Mitteilung). Math. Annalen, 96:169-175, 1926.
[70] Ph. Furtwängler. Über die simultane Approximation von Irrationalzahlen (Zweite Mitteilung). Math. Annalen, 99:71-83, 1928.
[71] K. M. Briggs and G. Álvarez. Scaling in a map of the two-torus. Experimental Mathematics, 9:301-307, 2000.
[72] Seung-hwan Kim and Stellan Ostlund. Renormalization of mappings of the two-torus. Phys. Rev. Lett., 55:1165-1168, 1985.
[73] Seung-hwan Kim and Stellan Ostlund. Simultaneous rational approximations in the study of dynamical systems. Phys. Rev. A, 34:3426-3434, 1986.
[74] Keith Martin Briggs. A torus map based on Jacobi's sn. Computers and Graphics, 19:451-453, 1995.
Keywords: graphics, torus
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[76] G. Szekeres. The N-dimensional approximation constant. Bull. Austral. Math. Soc., 29:119-125, 1984.
[77] G. Szekeres. Computer examination of the 2-dimensional simultaneous approximation constant. Ars Combinatoria, 19A:237-243, 1985.
[78] G. Szekeres. Search for the three dimensional approximation constant. In J. H. Loxton and A. J. van der Poorten, editors, Diophantine analysis: proceedings of the Number Theory Section of the 1985 Australian Mathematical Society Convention, volume 109 of London Mathematical Society lecture note series, pages 139-146, 1986.
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[84] W. W. Adams. Some computations in diophantine approximations. J. reine angew. Math., 220:163-173, 1965. MR 32 #91.
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