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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,
. 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).

## Abstract

*α*be an arbitrary real root of a quadratic equation with integer coefficients; say,

*α*=

*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). 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.