Randomness of the square root of 2 and the Giant Leap, Part 1
Randomness of the square root of 2 and the Giant Leap, Part 1
复制标题
2 的平方根和大跃进的随机性，第 1 部分
DOI：
--
复制
发表时间：
2010
影响因子：
0.8
通讯作者：
J. Beck
中科院分区：
文献类型：
--
作者：
J. Beck
摘要我们证明了“二次无理旋转”具有中心极限定理。更准确地说，设α是具有整数系数的二次方程的任意实根；比方说，α=%MathType！MTEF！2！1！+-
%feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9
%gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav
%P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC
%0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq
%aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe
%qaamaaeaqbaaGcbaWaaOaaaeaacqaIYaGmaSqabaaaaa！3CE4！
$$
SQRT 2
$$。给定任何有理数0<；x<；1(比方说，x=1/2)和任何正整数n，我们计算序列α，2α，3α，…的元素数，nα模1落入子区间[0，x]。我们证明了这个计数在以下意义上满足一个中心极限定理。首先，我们从计数数中减去“预期数”Nx，并研究当n在长区间1≤n≤N中运行时，这种差异的典型波动。根据α和x，我们可能需要对N的常量乘以对数的额外的相加校正；此外，我们总是需要乘法校正：除以(另一个)常量乘以N的对数的平方根。如果N很大，当n运行在1≤n≤N中时，这个重整化计数数的分布非常接近于标准正态分布(钟形曲线)，并且随着N趋于无穷，相应的误差项趋于零。这是本文的主要结果(见定理1.1)。这个证明相当复杂和冗长；它有许多有趣的弯路和副产品。例如，精确确定关键常量因子(在加法和乘法赋范中)依赖于α和x，需要惊人地深入的代数工具，如Dedeking和、二次域的类数和广义类数公式。二次无理数的关键性质是它的连分式的周期性。周期性意味着自相似，这将我们引向马尔可夫链：我们证明中心极限定理的基本概率工具。我们还使用了大量的傅立叶分析。最后，我只提到这项研究的一个副产品：我们解决了Hardy和Littlewood关于丢番图和的一个老问题。全文由导言和17个部分组成。第1部分包括导言和第1-7节。
AbstractWe 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 = % MathType!MTEF!2!1!+-
% feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9
% gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav
% P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC
% 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq
% aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe
% qaamaaeaqbaaGcbaWaaOaaaeaacqaIYaGmaSqabaaaaa!3CE4!
$$
sqrt 2
$$. 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, 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.