# 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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