On Schmidt and Summerer parametric geometry of numbers

Abstract

Recently, W. M. Schmidt and L. Summerer introduced a new theory that allowed them to recover the main known inequalities relating the usual exponents of Diophantine approximation to a point in $\mathbb{R}^n$ and to discover new ones. They first note that these exponents can be computed in terms of the successive minima of a parametric family of convex bodies attached to the given point. Then they prove that the $n$-tuple of these successive minima can in turn be approximated up to bounded difference by a function from a certain class. In this paper, we show that the same is true within a smaller and simpler class of functions, which we call rigid systems. We also show that conversely, given a rigid system, there exists a point in $\mathbb{R}^n$ whose associated family of convex bodies has successive minima that approximate that rigid system up to bounded difference. As a consequence, the problem of describing the joint spectrum of a family of exponents of Diophantine approximation is reduced to combinatorial analysis.

  • [BL2010] Go to document Y. Bugeaud and M. Laurent, "On transfer inequalities in Diophantine approximation. II," Math. Z., vol. 265, iss. 2, pp. 249-262, 2010.
    @article{BL2010, mrkey = {2609309},
      author = {Bugeaud, Yann and Laurent, Michel},
      title = {On transfer inequalities in {D}iophantine approximation. {II}},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {265},
      year = {2010},
      number = {2},
      pages = {249--262},
      issn = {0025-5874},
      coden = {MAZEAX},
      mrclass = {11J13 (11J25 11K60)},
      mrnumber = {2609309},
      mrreviewer = {Serge Perrine},
      doi = {10.1007/s00209-009-0512-0},
      zblnumber = {1234.11086},
      }
  • [Ge2012] Go to document O. N. German, "Intermediate Diophantine exponents and parametric geometry of numbers," Acta Arith., vol. 154, iss. 1, pp. 79-101, 2012.
    @article{Ge2012, mrkey = {2943676},
      author = {German, Oleg N.},
      title = {Intermediate {D}iophantine exponents and parametric geometry of numbers},
      journal = {Acta Arith.},
      fjournal = {Acta Arithmetica},
      volume = {154},
      year = {2012},
      number = {1},
      pages = {79--101},
      issn = {0065-1036},
      coden = {AARIA9},
      mrclass = {11J13 (11H06 11J82)},
      mrnumber = {2943676},
      mrreviewer = {Michel Laurent},
      doi = {10.4064/aa154-1-5},
      zblnumber = {06043109},
      }
  • [GL1987] P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, Second ed., Amsterdam: North-Holland Publishing Co., 1987, vol. 37.
    @book{GL1987, mrkey = {0893813},
      author = {Gruber, P. M. and Lekkerkerker, C. G.},
      title = {Geometry of Numbers},
      series = {North-Holland Math. Library},
      volume = {37},
      edition = {Second},
      publisher = {North-Holland Publishing Co.},
      address = {Amsterdam},
      year = {1987},
      pages = {xvi+732},
      isbn = {0-444-70152-4},
      mrclass = {11Hxx (52-02)},
      mrnumber = {0893813},
      mrreviewer = {Thomas W. Cusick},
      zblnumber = {0611.10017},
      }
  • [Ja1938] V. Jarn’ik, "Zum Khintchineschen “Übertragungssatz”," Trav. Inst. Math. Tbilissi, vol. 3, pp. 193-212, 1938.
    @article{Ja1938,
      author = {Jarn\'ık, V.},
      title = {Zum {K}hintchineschen ``{{Ü}bertragungssatz}''},
      journal = {Trav. Inst. Math. Tbilissi},
      volume = {3},
      year = {1938},
      pages = {193--212},
      zblnumber = {0019.10602},
      }
  • [Kh1926a] Go to document A. Khintchine, "Zur metrischen Theorie der diophantischen Approximationen," Math. Z., vol. 24, iss. 1, pp. 706-714, 1926.
    @article{Kh1926a, mrkey = {1544787},
      author = {Khintchine, A.},
      title = {Zur metrischen {T}heorie der diophantischen {A}pproximationen},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {24},
      year = {1926},
      number = {1},
      pages = {706--714},
      issn = {0025-5874},
      coden = {MAZEAX},
      mrclass = {Contributed Item},
      mrnumber = {1544787},
      doi = {10.1007/BF01216806},
      JFMNUMBER = {52.0183.02},
     }
  • [Kh1926b] A. Khintchine, "Über eine Klasse linearer diophantischer Approximationen," Rend. Circ. Math. Palermo, vol. 50, pp. 170-195, 1926.
    @article{Kh1926b,
      author = {Khintchine, A.},
      title = {{Ü}ber eine {K}lasse linearer diophantischer {A}pproximationen},
      journal = {Rend. Circ. Math. Palermo},
      volume = {50},
      year = {1926},
      pages = {170--195},
      jfmnumber = {52.0183.01},
      }
  • [La2009] Go to document M. Laurent, "Exponents of Diophantine approximation in dimension two," Canad. J. Math., vol. 61, iss. 1, pp. 165-189, 2009.
    @article{La2009, mrkey = {2488454},
      author = {Laurent, Michel},
      title = {Exponents of {D}iophantine approximation in dimension two},
      journal = {Canad. J. Math.},
      fjournal = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
      volume = {61},
      year = {2009},
      number = {1},
      pages = {165--189},
      issn = {0008-414X},
      coden = {CJMAAB},
      mrclass = {11J13 (11J25 11J70)},
      mrnumber = {2488454},
      mrreviewer = {Nicolas Chevallier},
      doi = {10.4153/CJM-2009-008-2},
      zblnumber = {1229.11101},
      }
  • [La2009b] M. Laurent, "On transfer inequalities in Diophantine approximation," in Analytic Number Theory, Cambridge: Cambridge Univ. Press, 2009, pp. 306-314.
    @incollection{La2009b, mrkey = {2508652},
      author = {Laurent, Michel},
      title = {On transfer inequalities in {D}iophantine approximation},
      booktitle = {Analytic Number Theory},
      pages = {306--314},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
      year = {2009},
      mrclass = {11J25 (11K60)},
      mrnumber = {2508652},
      mrreviewer = {Serge Perrine},
      zblnumber = {1163.11053},
      }
  • [Ma1955] Go to document K. Mahler, "On compound convex bodies. I.," Proc. London Math. Soc., vol. 5, pp. 358-379, 380, 1955.
    @article{Ma1955, mrkey = {0074460},
      author = {Mahler, Kurt},
      title = {On compound convex bodies. {I.}},
      journal = {Proc. London Math. Soc.},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {5},
      year = {1955},
      pages = {358--379, 380--384},
      issn = {0024-6115},
      mrclass = {10.3X},
      mrnumber = {0074460},
      mrreviewer = {P. Scherk},
      doi = {10.1112/plms/s3-5.3.358},
      zblnumber = {0065.28002},
      }
  • [Ma1955b] Go to document K. Mahler, "On compound convex bodies. II.," Proc. London Math. Soc., vol. 5, pp. 358-379, 380, 1955.
    @article{Ma1955b, mrkey = {0074461},
      author = {Mahler, Kurt},
      title = {On compound convex bodies. {II.}},
      journal = {Proc. London Math. Soc.},
      fjournal = {Proceedings of the London Mathematical Society. Third Series},
      volume = {5},
      year = {1955},
      pages = {358--379, 380--384},
      issn = {0024-6115},
      mrclass = {10.3X},
      mrnumber = {0074461},
      mrreviewer = {P. Scherk},
      doi = {10.1112/plms/s3-5.3.380},
      zblnumber = {0065.28002},
      }
  • [Mo2012a] Go to document N. G. Moshchevitin, "Proof of W. M. Schmidt’s conjecture concerning successive minima of a lattice," J. Lond. Math. Soc., vol. 86, iss. 1, pp. 129-151, 2012.
    @article{Mo2012a, mrkey = {2959298},
      author = {Moshchevitin, N. G.},
      title = {Proof of {W}. {M}. {S}chmidt's conjecture concerning successive minima of a lattice},
      journal = {J. Lond. Math. Soc.},
      fjournal = {Journal of the London Mathematical Society. Second Series},
      volume = {86},
      year = {2012},
      number = {1},
      pages = {129--151},
      issn = {0024-6107},
      mrclass = {11H06 (11J13)},
      mrnumber = {2959298},
      mrreviewer = {Art{ū}ras Dubickas},
      doi = {10.1112/jlms/jdr076},
      zblnumber = {06073812},
      }
  • [Mo2012b] Go to document N. Moshchevitin, "Exponents for three-dimensional simultaneous Diophantine approximations," Czechoslovak Math. J., vol. 62(137), iss. 1, pp. 127-137, 2012.
    @article{Mo2012b, mrkey = {2899740},
      author = {Moshchevitin, Nikolay},
      title = {Exponents for three-dimensional simultaneous {D}iophantine approximations},
      journal = {Czechoslovak Math. J.},
      fjournal = {Czechoslovak Mathematical Journal},
      volume = {62(137)},
      year = {2012},
      number = {1},
      pages = {127--137},
      issn = {0011-4642},
      coden = {CZMJAE},
      mrclass = {11J13},
      mrnumber = {2899740},
      mrreviewer = {Clemens Fuchs},
      doi = {10.1007/s10587-012-0001-1},
      zblnumber = {1249.11061},
      }
  • [R_preprint_MZ] D. Roy, Spectrum of the exponents of best rational approximation, 2014.
    @misc{R_preprint_MZ,
      author = {Roy, D.},
      title = {Spectrum of the exponents of best rational approximation},
      year = {2014},
      arxiv = {1410.1007},
      }
  • [R_preprint] D. Roy, Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer.
    @misc{R_preprint,
      author = {Roy, D.},
      title = {Construction of points realizing the regular systems of {W}olfgang {S}chmidt and {L}eonard {S}ummerer},
      note = {{\it J. Théor. Nombres Bordeaux},
      to appear},
      }
  • [Sc1967] Go to document W. M. Schmidt, "On heights of algebraic subspaces and diophantine approximations," Ann. of Math., vol. 85, pp. 430-472, 1967.
    @article{Sc1967, mrkey = {0213301},
      author = {Schmidt, Wolfgang M.},
      title = {On heights of algebraic subspaces and diophantine approximations},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {85},
      year = {1967},
      pages = {430--472},
      issn = {0003-486X},
      mrclass = {10.30},
      mrnumber = {0213301},
      mrreviewer = {J. W. S. Cassels},
      doi = {10.2307/1970352},
      zblnumber = {0152.03602},
      }
  • [Sc1982] W. M. Schmidt, "Open problems in Diophantine approximation," in Diophantine Approximations and Transcendental Numbers, Birkhäuser, Boston, 1983, vol. 31, pp. 271-287.
    @incollection{Sc1982, mrkey = {0702204},
      author = {Schmidt, W. M.},
      title = {Open problems in {D}iophantine approximation},
      booktitle = {Diophantine Approximations and Transcendental Numbers},
      venue = {{L}uminy, 1982},
      series = {Progr. Math.},
      volume = {31},
      pages = {271--287},
      publisher = {Birkhäuser, Boston},
      year = {1983},
      mrclass = {11Jxx},
      mrnumber = {0702204},
      mrreviewer = {C. G. Lekkerkerker},
      zblnumber = {0529.10032},
      }
  • [SS2009] Go to document W. M. Schmidt and L. Summerer, "Parametric geometry of numbers and applications," Acta Arith., vol. 140, iss. 1, pp. 67-91, 2009.
    @article{SS2009, mrkey = {2557854},
      author = {Schmidt, Wolfgang M. and Summerer, Leonhard},
      title = {Parametric geometry of numbers and applications},
      journal = {Acta Arith.},
      fjournal = {Acta Arithmetica},
      volume = {140},
      year = {2009},
      number = {1},
      pages = {67--91},
      issn = {0065-1036},
      coden = {AARIA9},
      mrclass = {11H06 (11J13)},
      mrnumber = {2557854},
      mrreviewer = {Yann Bugeaud},
      doi = {10.4064/aa140-1-5},
      zblnumber = {1236.11060},
      }
  • [SS2013a] Go to document W. M. Schmidt and L. Summerer, "Diophantine approximation and parametric geometry of numbers," Monatsh. Math., vol. 169, iss. 1, pp. 51-104, 2013.
    @article{SS2013a, mrkey = {3016519},
      author = {Schmidt, Wolfgang M. and Summerer, Leonhard},
      title = {Diophantine approximation and parametric geometry of numbers},
      journal = {Monatsh. Math.},
      fjournal = {Monatshefte für Mathematik},
      volume = {169},
      year = {2013},
      number = {1},
      pages = {51--104},
      issn = {0026-9255},
      mrclass = {11J13 (11H06)},
      mrnumber = {3016519},
      mrreviewer = {Serge Perrine},
      doi = {10.1007/s00605-012-0391-z},
      zblnumber = {1264.11056},
      }
  • [SS2013b] Go to document W. M. Schmidt and L. Summerer, "Simultaneous approximation to three numbers," Mosc. J. Comb. Number Theory, vol. 3, pp. 84-107, 2013.
    @article{SS2013b,
      author = {Schmidt, Wolfgang M. and Summerer, Leonhard},
      title = {Simultaneous approximation to three numbers},
      journal = {Mosc. J. Comb. Number Theory},
      volume = {3},
      year = {2013},
      pages = {84--107},
      zblnumber = {06315604},
      url = {http://mjcnt.phystech.edu/en/article.php?id=68},
      }

Authors

Damien Roy

University of Ottawa, Ottawa, Ontario, Canada