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Superirregularity

Part of the Lecture Notes in Mathematics book series (LNM,volume 2107)

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.

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Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4
Fig. 4.5
Fig. 4.6
Fig. 4.7
Fig. 4.8
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Fig. 4.11

Notes

  1. 1.

    For simplicity of notation, it is more convenient to restrict the second variable y.

  2. 2.

    This is a well-known fact from the reduction theory of binary quadratic forms. We omit the proof; see, for example, [31].

  3. 3.

    For a more detailed proof; see [23].

  4. 4.

    If β = 0, then the line is \(y = x/\sqrt{2}\).

  5. 5.

    Indeed, we have \(x_{1}^{2} - 2y_{1}^{2} = (x + 2y)^{2} - 2(x + y)^{2} = -(x^{2} - 2y^{2})\).

  6. 6.

    Here a and b are generic numbers.

  7. 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. 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. 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. 10.

    Again, we do not distinguish between positive and negative slopes.

  11. 11.

    In fact, the majority will do.

  12. 12.

    Here we skip a lot of technical details!

  13. 13.

    We do not distinguish between positive and negative slopes.

  14. 14.

    Note that the special case \(\alpha = \sqrt{2}\) was introduced in Sect. 4.1; see (4.28).

  15. 15.

    The reason behind this change is rotation-invariance. Theorems 3 and 12 are about translated copies, whereas Theorem 30 is about rotated and translated copies of the hyperbolic needle.

  16. 16.

    Note that 2j η is the length of the long side of a j-cell.

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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

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