# Randomness of the square root of 2 and the giant leap, part 2

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## Abstract

We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let *α* be an arbitrary real root of a quadratic equation with integer coefficients; say, \(\alpha = \sqrt 2\)
. Given any rational number 0 < *x* < 1 (say, *x* = 1/2) and any positive integer *n*, we count the number of elements of the sequence *α*, 2*α*, 3*α*, ..., *nα* modulo 1 that fall into the subinterval [0, *x*]. We prove that this counting number satisfies a central limit theorem in the following sense. First, we subtract the “expected number” *nx* from the counting number, and study the typical fluctuation of this difference as n runs in a long interval 1 ≤ *n* ≤ *N*. Depending on *α* and *x*, we may need an extra additive correction of constant times logarithm of *N*; furthermore, what we always need is a multiplicative correction: division by (another) constant times square root of logarithm of *N*. If *N* is large, the distribution of this renormalized counting number, as n runs in 1 ≤ *n* ≤ *N*, is very close to the standard normal distribution (bell shaped curve), and the corresponding error term tends to zero as *N* tends to infinity. This is the main result of the paper (see Theorem 1.1).

## Key words and phrases

lattice point counting in specified regions discrepancy irregularities of distribution distribution modulo 1 central limit theorem continued fractions diophantine inequalities inhomogeneous linear forms Dedekind sums## Mathematics subject classification numbers

11P21 11K38 11K06 60F05.## Preview

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## References

- [Be6]J. Beck,
*Inevitable Randomness in Discrete Mathematics*, University Lecture Series, Vol. 49, Amer. Math. Soc., 2009.Google Scholar - [El]
- [Ha-Li1]G. H. Hardy and J. Littlewood, The lattice-points of a right-angled triangle. I,
*Proc. London Math. Soc.*,**3**(1920), 15–36.Google Scholar - [Ha-Li2]G. H. Hardy and J. Littlewood, The lattice-points of a right-angled triangle. II,
*Abh. Math. Sem. Hamburg*,**1**(1922), 212–249.CrossRefGoogle Scholar - [Ha-Wr]G. H. Hardy and E. M. Wright,
*An introduction to the theory of numbers*, 5th edition, Clarendon Press, Oxford, 1979.MATHGoogle Scholar - [Ka]M. Kac, Probability methods in some problems of analysis and number theory,
*Bull. Amer. Math. Soc.*, 55 (1949), 641–665.MathSciNetMATHCrossRefGoogle Scholar - [Ke]Kesten, H., Uniform distribution mod 1,
*Ann. Math.*, 71 (1960), 445–471; Part II,*Acta Arithm.*,**7**(1961), 355–360.MathSciNetCrossRefGoogle Scholar - [Kn2]
- [La]
- [Os]A. Ostrowski, Bemerkungen zur Theorie der Diophantischen Approximationen. I,
*Abh. Hamburg Sem.*,**1**(1922), 77–99.CrossRefGoogle Scholar - [Shi]T. Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers,
*J. Fac. Sci. Univ. Tokyo*, 23 (1976), 393–417.MathSciNetMATHGoogle Scholar - [So2]Vera T. Sós, On the discrepancy of the sequence }
*nα*},*Colloq. Math. Soc. János Bolyai***13**, 1974, 359–367.Google Scholar - [Wo]
- [Za2]D. B. Zagier, On the values at negative integers of the zeta-function of a real quadratic field,
*Einseignement Math*. (2),**22**(1976), 55–95.MathSciNetGoogle Scholar - [Za3]D. B. Zagier, Valeurs des functions zeta des corps quadratiques reels aux entiers negatifs,
*Journées Arithmétiques de Caen, Asterisque*,**41–42**(1977), 135–151.MathSciNetGoogle Scholar