Abstract
Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, where the slope is a quadratic irrational. Motivated by this relationship, we carry out here a systematic study of point counting with respect to translated or congruent families of any given long and narrow hyperbolic region. First we discuss the important special case when the underlying point set is the set of integer lattice points in the plane and the slope of the given hyperbolic region is arbitrary but fixed; see Theorems 3–21. Then we switch to the general case of an arbitrary point set of density one in the plane, and study point counting with respect to congruent copies of a given hyperbolic region; see Theorem 30. The main results are about the extra large discrepancy that we call superirregularity. This means that there is always a translated/congruent copy of any given long and narrow hyperbolic region of large area, for which the actual number of points in the copy differs from the area as much as possible, i.e. the discrepancy is at least a constant multiple of the area. Our theorems demonstrate, in a quantitative sense, that in point counting with respect to translated/congruent copies of any long and narrow hyperbolic region, superirregularity is inevitable.
Keywords
- Integral Solution
- Integer Solution
- Iterate Logarithm
- Pigeonhole Principle
- Hyperbolic Region
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptions










Notes
- 1.
For simplicity of notation, it is more convenient to restrict the second variable y.
- 2.
This is a well-known fact from the reduction theory of binary quadratic forms. We omit the proof; see, for example, [31].
- 3.
For a more detailed proof; see [23].
- 4.
If β = 0, then the line is \(y = x/\sqrt{2}\).
- 5.
Indeed, we have \(x_{1}^{2} - 2y_{1}^{2} = (x + 2y)^{2} - 2(x + y)^{2} = -(x^{2} - 2y^{2})\).
- 6.
Here a and b are generic numbers.
- 7.
We also say that such a point is visible, explained by the geometric fact that the line segment with n and the origin as endpoints does not contain another lattice point. If \(\mathbf{n} = (n_{1},n_{2}) \in \mathbf{Z}^{2}\) were not coprime, then the point \((n_{1}/d,n_{2}/d) \in \mathbf{Z}^{2}\), where d ≥ 2 is the greatest common divisor of n 1 and n 2, would lie on this line segment.
- 8.
Halász used this method, among many other things, to give an elegant new proof of Schmidt’s well-known discrepancy theorem; see [27].
- 9.
We do not distinguish between positive and negative slopes. Note that the reflected hyperbolic needle − H γ (N) has two long arcs: the upper arc, which is increasing, and the lower arc, which is decreasing; here the lower arc is below the upper arc. When we say that P i − H γ (N) intersects \(\mathcal{C}\), then it always means that at least one of the two long arcs of P i − H γ (N) intersects \(\mathcal{C}\). For example, in the trivial error discussed at the end of this paragraph, the intersection comes from the upper arc.
- 10.
Again, we do not distinguish between positive and negative slopes.
- 11.
In fact, the majority will do.
- 12.
Here we skip a lot of technical details!
- 13.
We do not distinguish between positive and negative slopes.
- 14.
- 15.
- 16.
Note that 2j η is the length of the long side of a j-cell.
References
J. Beck, Randomness of \(n\sqrt{2}\) mod 1 and a Ramsey property of the hyperbola, in Sets, Graphs and Numbers, ed. by G. Halász, L. Lovász, D. Miklós, T. Szőnyi. Colloquia Math. Soc. János Bolyai, vol. 60 (North-Holland Publishing, Amsterdam, 1992), pp. 23–66
J. Beck, Randomness in diophantine approximation (Springer, to appear)
J. Beck, Diophantine approximation and quadratic fields, in Number Theory, ed. by K. Győry, A. Pethő, V.T. Sós. (Walter de Gruyter, Berlin, 1998), pp. 53–93
J. Beck, From probabilistic diophantine approximation to quadratic fields, in Random and Quasi-Random Point Sets, ed. by Hellekalek, P., Larcher G. Lecture Notes in Statistics, vol. 138 (Springer, New York, NY, 1998), pp. 1–48
J. Beck, Randomness in lattice point problems. Discrete Math. 229(1–3), 29–55 (2001). doi:10.1016/S0012-365X(00)00200-4
J. Beck, Lattice point problems: crossroads of number theory, probability theory and Fourier analysis, in Fourier Analysis and Convexity, ed. by L. Brandolini, L. Colzani, A. Iosevich, G. Travaglini (Birkhäuser, Boston, MA, 2004), pp. 1–35
J. Beck, Inevitable randomness in discrete mathematics. University Lecture Series, vol. 49 (American Mathematical Society (AMS), Providence, RI, 2009)
J. Beck, Lattice point counting and the probabilistic method. J. Combinator. 1(2), 171–232 (2010)
J. Beck, Randomness of the square root of 2 and the giant leap. I. Period. Math. Hung. 60(2), 137–242 (2010). doi:10.1007/s10998-010-2137-9
J. Beck, Randomness of the square root of 2 and the giant leap. II. Period. Math. Hung. 62(2), 127–246 (2011). doi:10.1007/s10998-011-6127-3
J. Beck, W.W.L. Chen, Irregularities of distribution. Cambridge Tracts in Mathematics vol. 89 (Cambridge University Press, Cambridge, 1987)
J.W.S. Cassels, An extension of the law of the iterated logarithm. Math. Proc. Cambridge Philos. Soc. 47, 55–64 (1951)
B. Chazelle, The discrepancy method. Randomness and complexity (Cambridge University Press, Cambridge, 2000)
P. Erdös, On the law of the iterated logarithm. Ann. Math. (2) 43, 419–436 (1942). doi:10.2307/1968801
W. Feller, The general form of the so-called law of the iterated logarithm. Trans. Am. Math. Soc. 54, 373–402 (1943). doi:10.2307/1990253
W. Feller, An introduction to probability theory and its applications. I., 3rd edn. (Wiley, New York, 1968)
W. Feller, An introduction to probability theory and its applications. II., 2nd edn. (Wiley, New York, 1971)
R.L. Graham, B.L. Rothschild, J.H. Spencer, Ramsey theory (Wiley, New York, 1980)
G. Halász, On Roth’s method in the theory of irregularities of point distributions, in Recent Progress in Analytic Number Theory, vol. 2, ed. by H. Halberstam, C. Hooley, vol. 2 (Academic Press, London, 1981), pp. 79–94
M. Kac, Probability methods in some problems of analysis and number theory. Bull. Am. Math. Soc. 55, 641–665 (1949). doi:10.1090/S0002-9904-1949-09242-X
A. Khintchine, Über einen Satz der Wahrscheinlichkeitsrechnung. Fund. math. 6, 9–20 (1924)
A. Kolmogorov, Über das Gesetz des iterierten Logarithmus. Math. Ann. 101, 126–135 (1929). doi:10.1007/BF01454828
S. Lang, Introduction to diophantine approximations (Addison-Wesley, Reading, 1966)
J. Matoušek, Geometric discrepancy. An illustrated guide. Revised paperback reprint of the 1999 original. Algorithms and Combinatorics, vol. 18 (Springer, Berlin, 2010). doi:10.1007/978-3-642-03942-3
A. Ostrowski, Bemerkungen zur Theorie der diophantischen Approximationen I. Abh. Math. Sem. Univ. Hamburg 1, 77–98 (1921). doi:10.1007/BF02940581
K.F. Roth, On irregularities of distribution. Mathematika 1, 73–79 (1954). doi:10.1112/S0025579300000541
W.M. Schmidt, Irregularities of distribution. VII. Acta Arith. 21, 45–50 (1972)
J.G. van der Corput, Verteilungsfunktionen. I. Proc. Kon. Ned. Akad. v. Wetensch. Amsterdam 38, 813–821 (1935)
J.G. van der Corput, Verteilungsfunktionen. II. Proc. Kon. Ned. Akad. v. Wetensch. Amsterdam 38, 1058–1066 (1935)
H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352 (1916). doi:10.1007/BF01475864
D.B. Zagier, Zetafunktionen und quadratische Körper. Eine Einführung in die höhere Zahlentheorie (Springer, Berlin, 1981)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Beck, J. (2014). Superirregularity. In: Chen, W., Srivastav, A., Travaglini, G. (eds) A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol 2107. Springer, Cham. https://doi.org/10.1007/978-3-319-04696-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-04696-9_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04695-2
Online ISBN: 978-3-319-04696-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
