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Relevant bibliographies by topics / Number theory – Diophantine approximation, transcendental number theory – Diophantine approximation, transcendental number theory / Journal articles

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Journal articles on the topic 'Number theory – Diophantine approximation, transcendental number theory – Diophantine approximation, transcendental number theory'

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Published: 4 June 2021

Last updated: 3 February 2022

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1

Kudin, Alexey, and Denis Vasilyev. "Counting real algebraic numbers with bounded derivative of minimal polynomial." International Journal of Number Theory 15, no. 10 (2019): 2223–39. http://dx.doi.org/10.1142/s1793042119501227.

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In this paper, we consider the problem of counting algebraic numbers [Formula: see text] of fixed degree [Formula: see text] and bounded height [Formula: see text] such that the derivative of the minimal polynomial [Formula: see text] of [Formula: see text] is bounded, [Formula: see text]. This problem has many applications to the problems of metric theory of Diophantine approximation. We prove that the number of [Formula: see text] defined above on the interval [Formula: see text] does not exceed [Formula: see text] for [Formula: see text] and [Formula: see text]. Our result is based on an im

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2

Muñoz Garcia, E., and R. Pérez-Marco. "Diophantine conditions in small divisors and transcendental number theory." Discrete & Continuous Dynamical Systems - A 9, no. 6 (2003): 1401–9. http://dx.doi.org/10.3934/dcds.2003.9.1401.

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3

Mundici, Daniele. "Triangles in diophantine approximation." Journal of Number Theory 201 (August 2019): 176–89. http://dx.doi.org/10.1016/j.jnt.2019.02.011.

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4

Florek, Jan. "Billiard and diophantine approximation." Acta Arithmetica 134, no. 4 (2008): 317–27. http://dx.doi.org/10.4064/aa134-4-2.

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5

Cusick, T. W., A. M. Rockett, and P. Szusz. "On Inhomogeneous Diophantine Approximation." Journal of Number Theory 48, no. 3 (1994): 259–83. http://dx.doi.org/10.1006/jnth.1994.1067.

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6

Tong, Jingcheng. "Diophantine approximation of a single irrational number." Journal of Number Theory 35, no. 1 (1990): 53–57. http://dx.doi.org/10.1016/0022-314x(90)90102-w.

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7

Schleischitz, Johannes. "Uniform Diophantine approximation and best approximation polynomials." Acta Arithmetica 185, no. 3 (2018): 249–74. http://dx.doi.org/10.4064/aa170901-4-7.

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8

Bugeaud, Yann, and Nicolas Chevallier. "On simultaneous inhomogeneous Diophantine approximation." Acta Arithmetica 123, no. 2 (2006): 97–123. http://dx.doi.org/10.4064/aa123-2-1.

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9

Nakamaye, Michael. "Diophantine approximation on algebraic varieties." Journal de Théorie des Nombres de Bordeaux 11, no. 2 (1999): 439–502. http://dx.doi.org/10.5802/jtnb.260.

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10

Pinner, Christopher G. "More on inhomogeneous diophantine approximation." Journal de Théorie des Nombres de Bordeaux 13, no. 2 (2001): 539–57. http://dx.doi.org/10.5802/jtnb.337.

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11

Fishman, Lior, David Simmons, and Mariusz Urbański. "Diophantine approximation in Banach spaces." Journal de Théorie des Nombres de Bordeaux 26, no. 2 (2014): 363–84. http://dx.doi.org/10.5802/jtnb.871.

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12

Dill, Gabriel A. "Effective approximation and Diophantine applications." Acta Arithmetica 177, no. 2 (2017): 169–99. http://dx.doi.org/10.4064/aa8430-9-2016.

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13

Vulakh, L. Y. "Diophantine Approximation On Bianchi Groups." Journal of Number Theory 54, no. 1 (1995): 73–80. http://dx.doi.org/10.1006/jnth.1995.1102.

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14

Ghosh, Anish. "Diophantine approximation on affine hyperplanes." Acta Arithmetica 144, no. 2 (2010): 167–82. http://dx.doi.org/10.4064/aa144-2-6.

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15

Bergelson, Vitaly, Inger J. H\aaland Knutson, and Randall McCutcheon. "Simultaneous Diophantine approximation and VIP systems." Acta Arithmetica 116, no. 1 (2005): 13–23. http://dx.doi.org/10.4064/aa116-1-2.

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16

Adiceam, Faustin. "Liminf Sets in Simultaneous Diophantine Approximation." Journal de Théorie des Nombres de Bordeaux 28, no. 2 (2016): 461–83. http://dx.doi.org/10.5802/jtnb.949.

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17

Urbański, Mariusz. "Diophantine approximation and self-conformal measures." Journal of Number Theory 110, no. 2 (2005): 219–35. http://dx.doi.org/10.1016/j.jnt.2004.07.004.

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18

Alkan, Emre, Glyn Harman, and Alexandru Zaharescu. "Diophantine approximation with mild divisibility constraints." Journal of Number Theory 118, no. 1 (2006): 1–14. http://dx.doi.org/10.1016/j.jnt.2005.08.001.

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19

Murty, M. Ram, and Hector Pasten. "Modular forms and effective Diophantine approximation." Journal of Number Theory 133, no. 11 (2013): 3739–54. http://dx.doi.org/10.1016/j.jnt.2013.05.006.

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20

Li, Taiyu, and Huafeng Liu. "Diophantine approximation over Piatetski-Shapiro primes." Journal of Number Theory 211 (June 2020): 184–98. http://dx.doi.org/10.1016/j.jnt.2019.10.002.

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21

Roy, Damien, and Michel Waldschmidt. "Diophantine approximation by conjugate algebraic integers." Compositio Mathematica 140, no. 03 (2004): 593–612. http://dx.doi.org/10.1112/s0010437x03000708.

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22

Ghosh, Anish, Alexander Gorodnik, and Amos Nevo. "Metric Diophantine approximation on homogeneous varieties." Compositio Mathematica 150, no. 8 (2014): 1435–56. http://dx.doi.org/10.1112/s0010437x13007859.

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AbstractWe develop the metric theory of Diophantine approximation on homogeneous varieties of semisimple algebraic groups and prove results analogous to the classical Khintchine and Jarník theorems. In full generality our results establish simultaneous Diophantine approximation with respect to several completions, and Diophantine approximation over general number fields using $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$-alge

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23

VULAKH, L. YA. "DIOPHANTINE APPROXIMATION IN IMAGINARY QUADRATIC FIELDS." International Journal of Number Theory 06, no. 04 (2010): 731–66. http://dx.doi.org/10.1142/s1793042110003137.

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Let H3 be the upper half-space model of the three-dimensional hyperbolic space. For certain cocompact Fuchsian subgroups Γ of an extended Bianchi group Bd, the extremality of the axis of hyperbolic F ∈ Γ in H3 with respect to Γ implies its extremality with respect to Bd. This reduction is used to obtain sharp lower bounds for the Hurwitz constants and lower bounds for the highest limit points in the Markov spectra of Bd for some d < 1000. In particular, such bounds are found for all non-Euclidean class one imaginary quadratic fields. The Hurwitz constants for the imaginary quadratic fields

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24

Marcovecchio, Raffaele. "Multiple Legendre polynomials in diophantine approximation." International Journal of Number Theory 10, no. 07 (2014): 1829–55. http://dx.doi.org/10.1142/s1793042114500584.

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We construct a class of multiple Legendre polynomials and prove that they satisfy an Apéry-like recurrence. We give new upper bounds of the approximation measures of logarithms of rational numbers by algebraic numbers of bounded degree. We prove, e.g., that the nonquadraticity exponent of log 2 is bounded from above by 12.841618…, thus improving upon a recent result of the author. Our construction also yields some other known results.

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25

Nesharim, Erez, René Rühr, and Ronggang Shi. "Metric Diophantine approximation with congruence conditions." International Journal of Number Theory 16, no. 09 (2020): 1923–33. http://dx.doi.org/10.1142/s1793042120500980.

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We prove a version of the Khinchin–Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.

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26

Tong, Jingcheng. "Segre's theorem on asymmetric diophantine approximation." Journal of Number Theory 28, no. 1 (1988): 116–18. http://dx.doi.org/10.1016/0022-314x(88)90122-9.

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27

Alam, Mahbub, Anish Ghosh, and Shucheng Yu. "Quantitative Diophantine approximation with congruence conditions." Journal de Théorie des Nombres de Bordeaux 33, no. 1 (2021): 261–71. http://dx.doi.org/10.5802/jtnb.1161.

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28

Baker, R., J. Brüder, and G. Harman. "Simultaneous diophantine approximation with square-free numbers." Acta Arithmetica 63, no. 1 (1993): 51–60. http://dx.doi.org/10.4064/aa-63-1-51-60.

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29

Inoue, Kae, and Hitoshi Nakada. "On metric Diophantine approximation in positive characteristic." Acta Arithmetica 110, no. 3 (2003): 205–18. http://dx.doi.org/10.4064/aa110-3-1.

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30

Bugeaud, Yann, and Bernard de Mathan. "On a mixed problem in Diophantine approximation." Acta Arithmetica 139, no. 1 (2009): 65–77. http://dx.doi.org/10.4064/aa139-1-6.

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31

Biró, András, and Vera T. Sós. "Strong characterizing sequences in simultaneous diophantine approximation." Journal of Number Theory 99, no. 2 (2003): 405–14. http://dx.doi.org/10.1016/s0022-314x(02)00068-9.

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32

Schmidt, Asmus L. "Diophantine approximation in the field Q(i2)." Journal of Number Theory 131, no. 10 (2011): 1983–2012. http://dx.doi.org/10.1016/j.jnt.2011.04.002.

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33

Liao, Lingmin, and Michał Rams. "Inhomogeneous Diophantine approximation with general error functions." Acta Arithmetica 160, no. 1 (2013): 25–35. http://dx.doi.org/10.4064/aa160-1-2.

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34

Roy, Damien. "On the topology of Diophantine approximation spectra." Compositio Mathematica 153, no. 7 (2017): 1512–46. http://dx.doi.org/10.1112/s0010437x17007126.

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Fix an integer$n\geqslant 2$. To each non-zero point$\mathbf{u}$in$\mathbb{R}^{n}$, one attaches several numbers calledexponents of Diophantine approximation. However, as Khintchine first observed, these numbers are not independent of each other. This raises the problem of describing the set of all possible values that a given family of exponents can take by varying the point $\mathbf{u}$. To avoid trivialities, one restricts to points $\mathbf{u}$whose coordinates are linearly independent over $\mathbb{Q}$. The resulting set of values is called thespectrum of these exponents. We show that, in

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35

Liu, Zhixin. "Diophantine approximation by unlike powers of primes." International Journal of Number Theory 13, no. 09 (2017): 2445–52. http://dx.doi.org/10.1142/s1793042117501330.

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Let [Formula: see text] be nonzero real numbers not all of the same sign, satisfying that [Formula: see text] is irrational, and [Formula: see text] be a real number. In this paper, we prove that for any [Formula: see text] [Formula: see text] has infinitely many solutions in prime variables [Formula: see text].

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36

Lagarias, Jeffrey C., and Johan T. Hastad. "Simultaneous Diophantine approximation of rationals by rationals." Journal of Number Theory 24, no. 2 (1986): 200–228. http://dx.doi.org/10.1016/0022-314x(86)90104-6.

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37

Dodson, M. M., B. P. Rynne, and J. A. G. Vickers. "Dirichlet's theorem and diophantine approximation on manifolds." Journal of Number Theory 36, no. 1 (1990): 85–88. http://dx.doi.org/10.1016/0022-314x(90)90006-d.

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38

Harman, Glyn. "Some theorems in the metric theory of diophantine approximation." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 3 (1986): 385–94. http://dx.doi.org/10.1017/s0305004100064331.

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An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequalityfor almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, and f is a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reas

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39

ZHANG, YUAN. "A NOTE ON MATRIX APPROXIMATION IN THE THEORY OF MULTIPLICATIVE DIOPHANTINE APPROXIMATION." Bulletin of the Australian Mathematical Society 100, no. 3 (2019): 372–77. http://dx.doi.org/10.1017/s000497271900039x.

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We prove the Hausdorff measure version of the matrix form of Gallagher’s theorem in the inhomogeneous setting, thereby proving a conjecture posed by Hussain and Simmons [‘The Hausdorff measure version of Gallagher’s theorem—closing the gap and beyond’, J. Number Theory186 (2018), 211–225].

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40

Harman, Glyn. "Simultaneous Diophantine approximation and asymptotic formulae on manifolds." Acta Arithmetica 108, no. 4 (2003): 379–89. http://dx.doi.org/10.4064/aa108-4-7.

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41

Bugeaud, Yann, Michael Drmota, and Bernard de Mathan. "On a mixed Littlewood conjecture in Diophantine approximation." Acta Arithmetica 128, no. 2 (2007): 107–24. http://dx.doi.org/10.4064/aa128-2-2.

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42

Rynne, Bryan P. "Simultaneous Diophantine approximation on manifolds and Hausdorff dimension." Journal of Number Theory 98, no. 1 (2003): 1–9. http://dx.doi.org/10.1016/s0022-314x(02)00035-5.

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43

Kaziulytė, Laima. "Variants of Khintchine's theorem in metric Diophantine approximation." Journal of Number Theory 215 (October 2020): 160–70. http://dx.doi.org/10.1016/j.jnt.2020.01.005.

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44

Dodson, M. M., B. P. Rynne, and J. A. G. Vickers. "Simultaneous Diophantine Approximation and Asymptotic Formulae on Manifolds." Journal of Number Theory 58, no. 2 (1996): 298–316. http://dx.doi.org/10.1006/jnth.1996.0079.

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45

Ru, Min, and Julie Tzu-Yueh Wang. "Diophantine Approximation with Algebraic Points of Bounded Degree." Journal of Number Theory 81, no. 1 (2000): 110–19. http://dx.doi.org/10.1006/jnth.1999.2462.

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46

Yu, Han. "A Fourier-analytic approach to inhomogeneous Diophantine approximation." Acta Arithmetica 190, no. 3 (2019): 263–92. http://dx.doi.org/10.4064/aa180627-25-9.

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47

Qu, Yunyun, and Jiwen Zeng. "Diophantine approximation with prime variables and mixed powers." Ramanujan Journal 52, no. 3 (2019): 625–39. http://dx.doi.org/10.1007/s11139-019-00167-8.

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48

Chow, Sam, and Niclas Technau. "Higher-rank Bohr sets and multiplicative diophantine approximation." Compositio Mathematica 155, no. 11 (2019): 2214–33. http://dx.doi.org/10.1112/s0010437x19007589.

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Gallagher’s theorem is a sharpening and extension of the Littlewood conjecture that holds for almost all tuples of real numbers. We provide a fibre refinement, solving a problem posed by Beresnevich, Haynes and Velani in 2015. Hitherto, this was only known on the plane, as previous approaches relied heavily on the theory of continued fractions. Using reduced successive minima in lieu of continued fractions, we develop the structural theory of Bohr sets of arbitrary rank, in the context of diophantine approximation. In addition, we generalise the theory and result to the inhomogeneous setting.

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49

Xiong, Maosheng, and Alexandru Zaharescu. "A problem of Erdős–Szüsz–Turán on diophantine approximation." Acta Arithmetica 125, no. 2 (2006): 163–77. http://dx.doi.org/10.4064/aa125-2-3.

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50

Zhang, Yuqing. "Diophantine exponents of affine subspaces: The simultaneous approximation case." Journal of Number Theory 129, no. 8 (2009): 1976–89. http://dx.doi.org/10.1016/j.jnt.2009.02.013.

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