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Equidistribution theorem
Equidistribution theorem
Equidistribution theorem
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Illustration of filling the unit interval (horizontal axis) with the first n terms using the equidistribution theorem with four common irrational numbers, for n from 0 to 999 (vertical axis). The 113 distinct bands for π are due to the closeness of its value to the rational number 355/113. Similarly, the 7 distinct groups are due to π being approximately 22/7.
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In mathematics, the equidistribution theorem is the statement that the sequence

a, 2a, 3a, ... mod 1

is uniformly distributed on the circle , when a is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure .

History

[edit]

While this theorem was proved in 1909 and 1910 separately by Hermann Weyl, Wacław Sierpiński and Piers Bohl, variants of this theorem continue to be studied to this day.

In 1916, Weyl proved that the sequence a, 22a, 32a, ... mod 1 is uniformly distributed on the unit interval. In 1937, Ivan Vinogradov proved that the sequence pn a mod 1 is uniformly distributed, where pn is the nth prime. Vinogradov's proof was a byproduct of the odd Goldbach conjecture, that every sufficiently large odd number is the sum of three primes.

George Birkhoff, in 1931, and Aleksandr Khinchin, in 1933, proved that the generalization x + na, for almost all x, is equidistributed on any Lebesgue measurable subset of the unit interval. The corresponding generalizations for the Weyl and Vinogradov results were proven by Jean Bourgain in 1988.

Specifically, Khinchin showed that the identity

holds for almost all x and any Lebesgue integrable function ƒ. In modern formulations, it is asked under what conditions the identity

might hold, given some general sequence bk.

One noteworthy result is that the sequence 2ka mod 1 is uniformly distributed for almost all, but not all, irrational a. Similarly, for the sequence bk = 2ka, for every irrational a, and almost all x, there exists a function ƒ for which the sum diverges. In this sense, this sequence is considered to be a universally bad averaging sequence, as opposed to bk = k, which is termed a universally good averaging sequence, because it does not have the latter shortcoming.

A powerful general result is Weyl's criterion, which shows that equidistribution is equivalent to having a non-trivial estimate for the exponential sums formed with the sequence as exponents. For the case of multiples of a, Weyl's criterion reduces the problem to summing finite geometric series.

See also

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References

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Equidistribution theorem

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The equidistribution theorem, often referred to as Weyl's equidistribution theorem, asserts that if α\alpha is an irrational real number, then the sequence of fractional parts {nα}\{n\alpha\} for positive integers nn is equidistributed in the unit interval [0,1)[0,1), meaning that for any subinterval [a,b]⊆[0,1)[a,b] \subseteq [0,1), the proportion of the first NN terms falling into [a,b][a,b] approaches b−ab-a as N→∞N \to \infty. This uniform distribution implies that the sequence densely fills the interval without clustering or gaps in the limit, distinguishing equidistribution from mere density. The basic equidistribution theorem for linear sequences was independently established in 1909–1910 by the Latvian mathematician Piers Bohl, the German mathematician Hermann Weyl, and the Polish mathematician Wacław Sierpiński. Weyl further developed the general theory of uniform distribution modulo one in 1916, building on earlier ideas in Diophantine approximation and Fourier analysis, as part of his work on uniform distribution in manifolds. Weyl's proof utilized exponential sums to demonstrate the uniformity, revealing deep connections between irrationality measures and asymptotic behavior of sequences. Prior contributions, such as those by Hardy and Littlewood in 1914 on sequences like {xn}\{x^n\} for almost all x>1x > 1, laid groundwork but did not fully generalize to linear sequences modulo 1. A cornerstone of the theory is Weyl's criterion, which provides a frequency-domain characterization: a sequence in [0,1)[0,1) is equidistributed if and only if the average of e2πikxne^{2\pi i k x_n} over the first NN terms tends to 0 as N→∞N \to \infty for every nonzero integer kk. This equivalence leverages the completeness of trigonometric polynomials in the space of continuous functions on the torus, enabling proofs via bounding discrepancies in exponential sums. The criterion extends naturally to higher dimensions, where equidistribution in the unit cube [0,1)d[0,1)^d requires the averages to vanish for all nonzero integer vectors. Beyond its classical form, the theorem has been generalized to polynomial sequences p(n)p(n) modulo 1, where equidistribution holds if at least one coefficient other than the constant term is irrational, impacting fields like ergodic theory and analytic number theory. Applications include estimating averages of periodic functions over sequences, approximating integrals, and studying discrepancy in pseudorandom number generation.

Foundations

Uniform distribution modulo one

A sequence {xn}n=1∞\{x_n\}_{n=1}^\infty of real numbers is said to be uniformly distributed modulo 1 if, for every subinterval [a,b)⊆[0,1)[a, b) \subseteq [0, 1) with 0≤a<b≤10 \leq a < b \leq 1, the proportion of the first NN terms falling into that subinterval approaches the length of the interval as N→∞N \to \infty. Formally, this means lim⁡N→∞1N∑k=1N1[a,b)({xk})=b−a,\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N \mathbf{1}_{[a, b)}(\{x_k\}) = b - a, where {xk}=xk−⌊xk⌋\{x_k\} = x_k - \lfloor x_k \rfloor denotes the fractional part of xkx_k, and 1[a,b)\mathbf{1}_{[a, b)} is the indicator function of the interval [a,b)[a, b). This condition can be interpreted in terms of the empirical measure associated with the sequence. The empirical measure μN\mu_N is defined as μN=1N∑k=1Nδ{xk}\mu_N = \frac{1}{N} \sum_{k=1}^N \delta_{\{x_k\}}, where δy\delta_y is the Dirac delta at yy. Uniform distribution modulo 1 holds if μN\mu_N converges weakly to the Lebesgue measure λ\lambda on [0,1)[0, 1) as N→∞N \to \infty, meaning that for any continuous function f:[0,1)→Rf: [0, 1) \to \mathbb{R}, lim⁡N→∞1N∑k=1Nf({xk})=∫01f(x) dλ(x)=∫01f(x) dx.\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N f(\{x_k\}) = \int_0^1 f(x) \, d\lambda(x) = \int_0^1 f(x) \, dx. This convergence captures the idea that the sequence spreads out evenly across the unit interval, mimicking the uniform probability distribution. The unit interval [0,1)[0, 1) with the endpoints identified forms the one-dimensional torus T\mathbb{T}, and the Lebesgue measure λ\lambda is the unique translation-invariant probability measure on T\mathbb{T}. Uniform distribution modulo 1 thus describes sequences whose fractional parts become asymptotically equidistributed with respect to this Haar measure on the torus. A fundamental property is that any uniformly distributed sequence is dense in [0,1)[0, 1), as the even spreading prevents accumulation in any proper subinterval. The concept extends naturally to multidimensional settings. A sequence in [0,1)d[0, 1)^d is uniformly distributed modulo 1 if its empirical measure converges to the dd-dimensional Lebesgue measure, which is the product measure λ⊗d\lambda^{\otimes d} on the dd-torus Td\mathbb{T}^d. This generalization preserves the density property in the higher-dimensional unit cube.

Weyl's criterion

Weyl's criterion provides a Fourier-analytic characterization of equidistribution modulo 1 for a sequence {xn}\{x_n\} of real numbers. It states that the sequence is equidistributed in [0,1)[0,1) if and only if, for every nonzero integer m∈Z∖{0}m \in \mathbb{Z} \setminus \{0\}, lim⁡N→∞1N∑k=1Ne2πimxk=0.\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N e^{2\pi i m x_k} = 0. The case m=0m=0 is trivial, as the exponential sum reduces to the average of 1, which always approaches 1 regardless of equidistribution. This criterion reduces the problem of verifying equidistribution—originally a question in measure theory—to estimating exponential sums, a tool from harmonic analysis that has proven powerful for applications in number theory. The derivation of Weyl's criterion relies on the orthogonality of characters on the circle group T=R/Z\mathbb{T} = \mathbb{R}/\mathbb{Z}. The functions χm(x)=e2πimx\chi_m(x) = e^{2\pi i m x} for m∈Zm \in \mathbb{Z} form a complete orthonormal basis for L2([0,1))L^2([0,1)) with respect to the Lebesgue measure. A sequence is equidistributed modulo 1 if its empirical measures μN=1N∑k=1Nδ{xk}\mu_N = \frac{1}{N} \sum_{k=1}^N \delta_{ \{x_k\} } converge weakly to the Lebesgue measure λ\lambda on [0,1)[0,1). Weak convergence implies that for any continuous function ff, ∫f dμN→∫f dλ\int f \, d\mu_N \to \int f \, d\lambda. Since the characters χm\chi_m are continuous and their Fourier coefficients determine the measure, the condition follows: for m≠0m \neq 0, ∫χm dλ=0\int \chi_m \, d\lambda = 0, so the averages must vanish; conversely, if the averages vanish for all m≠0m \neq 0, then by density of trigonometric polynomials in the continuous functions (via Fejér's theorem or Stone-Weierstrass), the convergence holds for all continuous ff. This criterion extends naturally to the multidimensional setting. For a sequence of vectors xn=(xn(1),…,xn(d))∈[0,1)d\mathbf{x}_n = (x_n^{(1)}, \dots, x_n^{(d)}) \in [0,1)^d, equidistribution holds if and only if, for every nonzero lattice point h=(h1,…,hd)∈Zd∖{0}\mathbf{h} = (h_1, \dots, h_d) \in \mathbb{Z}^d \setminus \{\mathbf{0}\}, lim⁡N→∞1N∑k=1Ne2πi⟨h,xk⟩=0,\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N e^{2\pi i \langle \mathbf{h}, \mathbf{x}_k \rangle} = 0, where ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle denotes the standard inner product. The derivation parallels the one-dimensional case, using the orthogonality of characters on the dd-torus Td\mathbb{T}^d.

Statement and proof

Formal statement

The equidistribution theorem addresses the uniform distribution of sequences modulo 1, beginning with the linear case. If α∈R\alpha \in \mathbb{R} is irrational, then the sequence of fractional parts {nα}n=1∞\{n\alpha\}_{n=1}^\infty, where {x}=x−⌊x⌋\{x\} = x - \lfloor x \rfloor denotes the fractional part of xx, is equidistributed in the unit interval [0,1)[0,1). This means that for any subinterval [a,b)⊆[0,1)[a,b) \subseteq [0,1), the proportion of terms {nα}\{n\alpha\} falling in [a,b)[a,b) up to NN approaches b−ab-a as N→∞N \to \infty. Weyl generalized this to polynomial sequences. Consider a real polynomial P(t)=adtd+⋯+a1t+a0P(t) = a_d t^d + \cdots + a_1 t + a_0 of degree d≥1d \geq 1. The sequence {P(n)}n=1∞\{P(n)\}_{n=1}^\infty is equidistributed modulo 1 if and only if at least one coefficient aja_j with 1≤j≤d1 \leq j \leq d is irrational. Equivalently, the irrationality condition ensures that the sequence does not concentrate on a finite union of arithmetic progressions modulo 1. This criterion can be verified using Weyl's equidistribution criterion, which equates equidistribution to the vanishing of certain exponential sums. In the multidimensional setting, the theorem extends to polynomial maps P:Z→Rm/ZmP: \mathbb{Z} \to \mathbb{R}^m / \mathbb{Z}^m, where P(n)=(P1(n),…,Pm(n))P(n) = (P_1(n), \dots, P_m(n)) and each PiP_i is a real polynomial. The sequence {P(n)}n=1∞\{P(n)\}_{n=1}^\infty is equidistributed in the unit cube [0,1)m[0,1)^m if the leading coefficients of the PiP_i (those of the highest degree terms) generate a subspace that is irrational with respect to the rational numbers, meaning their joint values on integer inputs span a full-dimensional irrational extension rather than lying in a proper rational subspace. This ensures the sequence densely and uniformly fills the torus without bias toward rational sublattices.

Proof outline

The proof of the equidistribution theorem relies on Weyl's criterion, which reduces the equidistribution of the sequence {P(n)}\{P(n)\} modulo one to verifying that the exponential sums vanish in the limit: for every integer m≠0m \neq 0, lim⁡N→∞1N∑k=1Ne2πimP(k)=0.\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N e^{2\pi i m P(k)} = 0. This criterion transforms the problem into estimating these trigonometric sums and showing their sublinear growth relative to NN. In the linear case where P(k)=αkP(k) = \alpha k and α\alpha is irrational, the sum is a geometric series: ∣∑k=1Ne2πimαk∣=∣sin⁡(πNmα)sin⁡(πmα)∣≤1∣sin⁡(πmα)∣.\left| \sum_{k=1}^N e^{2\pi i m \alpha k} \right| = \left| \frac{\sin(\pi N m \alpha)}{\sin(\pi m \alpha)} \right| \leq \frac{1}{|\sin(\pi m \alpha)|}.
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