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SIMULTANEOUS APPROXIMATION TO TWO REALS: BOUNDS FOR THE SECOND SUCCESSIVE MINIMUM

  • Wolfgang M. Schmidt (a1) and Leonhard Summerer (a2)
Abstract

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith. 140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$ , $(i=1,2,3)$ , attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$ , give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math. 61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “ $3$ -systems”.

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References
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1. Laurent, M., Exponents of Diophantine approximation in dimension two. Canad. J. Math. 61 2009, 165189.
2. Roy, D., On Schmidt and Summerer parametric geometry of numbers. Ann. of Math. (2) 182 2015, 739786.
3. Roy, D., On the topology of Diophantine approximation spectra. Compos. Math. 153 2017, 15121546.
4. Schmidt, W. M. and Summerer, L., Parametric geometry of numbers and applications. Acta Arith. 140 2009, 6791.
5. Schmidt, W. M. and Summerer, L., Diophantine approximation and parametric geometry of numbers. Monatsh. Math. 169 2013, 51104.
6. Schmidt, W. M. and Summerer, L., The generalization of Jarnik’s identity. Acta Arith. 175 2016, 119136.
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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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