RANDOMNESS OF THE SQUARE ROOT OF 2 AND THE GIANT LEAP, PART 2

作者：Beck Jozsef^*

来源：Periodica Mathematica Hungarica, 2011, 62(2): 127-246.

DOI：10.1007/s10998-011-6127-3

• 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

We prove that the "quadratic irrational rotation" exhibits a central limit theorem. More precisely, let alpha be an arbitrary real root of a quadratic equation with integer coefficients; say, alpha = root 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 alpha, 2 alpha, 3 alpha, ... , n alpha 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 alpha 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).

• 出版日期2011-6