# Randomness of the square root of 2 and the Giant Leap, Part 1

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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, α = √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, 2α, 3α..., 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). The proof is rather complicated and long; it has many interesting detours and byproducts. For example, the exact determination of the key constant factors (in the additive and multiplicative norming), which depend on α and x, requires surprisingly deep algebraic tools such as Dedeking sums, the class number of quadratic fields, and generalized class number formulas. The crucial property of a quadratic irrational is the periodicity of its continued fraction. Periodicity means self-similarity, which leads us to Markov chains: our basic probabilistic tool to prove the central limit theorem. We also use a lot of Fourier analysis. Finally, I just mention one byproduct of this research: we solve an old problem of Hardy and Littlewood on diophantine sums. The whole paper consists of an introduction and 17 sections. Part 1 contains the Introduction and Sections 1-7.

Original language English (US) 137-242 106 Periodica Mathematica Hungarica 60 2 https://doi.org/10.1007/s10998-010-2137-9 Published - Jun 1 2010

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Keywords

• Central limit theorem
• Continued fractions
• Dedekind sums
• Diophantine inequalities
• Discrepancy
• Distribution modulo 1
• Inhomogeneous linear forms
• Irregularities of distribution
• Lattice point counting in specified regions