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Equidistribution theorem

The equidistribution theorem, often referred to as Weyl's equidistribution theorem, asserts that if \alpha is an irrational real number, then the sequence of fractional parts \{n\alpha\} for positive integers n is equidistributed in the unit interval [0,1), meaning that for any subinterval [a,b] \subseteq [0,1), the proportion of the first N terms falling into [a,b] approaches b-a as N \to \infty.^[1]^[2] This uniform distribution implies that the sequence densely fills the interval without clustering or gaps in the limit, distinguishing equidistribution from mere density.^[1] 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.^[3] 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.^[4] 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 \{x^n\} for almost all x > 1, laid groundwork but did not fully generalize to linear sequences modulo 1.^[1] A cornerstone of the theory is Weyl's criterion, which provides a frequency-domain characterization: a sequence in [0,1) is equidistributed if and only if the average of e^{2\pi i k x_n} over the first N terms tends to 0 as N \to \infty for every nonzero integer k. 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.^[2] The criterion extends naturally to higher dimensions, where equidistribution in the unit cube [0,1)^d requires the averages to vanish for all nonzero integer vectors.^[4] Beyond its classical form, the theorem has been generalized to polynomial sequences 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.^[4]

Foundations

Uniform distribution modulo one

A sequence \{x_n\}_{n=1}^\infty of real numbers is said to be uniformly distributed modulo 1 if, for every subinterval [a, b) \subseteq [0, 1) with $0 \leq a < b \leq 1, the proportion of the first N terms falling into that subinterval approaches the length of the interval as N \to \infty. Formally, this means

\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N \mathbf{1}_{[a, b)}(\{x_k\}) = b - a,

where \{x_k\} = x_k - \lfloor x_k \rfloor denotes the fractional part of x_k, and \mathbf{1}_{[a, b)} is the indicator function of the interval [a, b).^[5]^[6] This condition can be interpreted in terms of the empirical measure associated with the sequence. The empirical measure \mu_N is defined as \mu_N = \frac{1}{N} \sum_{k=1}^N \delta_{\{x_k\}}, where \delta_y is the Dirac delta at y. Uniform distribution modulo 1 holds if \mu_N converges weakly to the Lebesgue measure \lambda on [0, 1) as N \to \infty, meaning that for any continuous function f: [0, 1) \to \mathbb{R},

\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.^[5]^[7] The unit interval [0, 1) with the endpoints identified forms the one-dimensional torus \mathbb{T}, and the Lebesgue measure \lambda is the unique translation-invariant probability measure on \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), as the even spreading prevents accumulation in any proper subinterval.^[5]^[8] The concept extends naturally to multidimensional settings. A sequence in [0, 1)^d is uniformly distributed modulo 1 if its empirical measure converges to the d-dimensional Lebesgue measure, which is the product measure \lambda^{\otimes d} on the d-torus \mathbb{T}^d. This generalization preserves the density property in the higher-dimensional unit cube.^[5]^[6]

Weyl's criterion

Weyl's criterion provides a Fourier-analytic characterization of equidistribution modulo 1 for a sequence \{x_n\} of real numbers. It states that the sequence is equidistributed in [0,1) if and only if, for every nonzero integer m \in \mathbb{Z} \setminus \{0\},

\lim_{N \to \infty} \frac{1}{N} \sum_{k=1}^N e^{2\pi i m x_k} = 0.

The case m=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.^[9]^[5] The derivation of Weyl's criterion relies on the orthogonality of characters on the circle group \mathbb{T} = \mathbb{R}/\mathbb{Z}. The functions \chi_m(x) = e^{2\pi i m x} for m \in \mathbb{Z} form a complete orthonormal basis for L^2([0,1)) with respect to the Lebesgue measure. A sequence is equidistributed modulo 1 if its empirical measures \mu_N = \frac{1}{N} \sum_{k=1}^N \delta_{ \{x_k\} } converge weakly to the Lebesgue measure \lambda on [0,1). Weak convergence implies that for any continuous function f, \int f \, d\mu_N \to \int f \, d\lambda. Since the characters \chi_m are continuous and their Fourier coefficients determine the measure, the condition follows: for m \neq 0, \int \chi_m \, d\lambda = 0, so the averages must vanish; conversely, if the averages vanish for all m \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 f.^[5]^[10] This criterion extends naturally to the multidimensional setting. For a sequence of vectors \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 \mathbf{h} = (h_1, \dots, h_d) \in \mathbb{Z}^d \setminus \{\mathbf{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 d-torus \mathbb{T}^d.^[5]

Statement and proof

Formal statement

The equidistribution theorem addresses the uniform distribution of sequences modulo 1, beginning with the linear case. If \alpha \in \mathbb{R} is irrational, then the sequence of fractional parts \{n\alpha\}_{n=1}^\infty, where \{x\} = x - \lfloor x \rfloor denotes the fractional part of x, is equidistributed in the unit interval [0,1). This means that for any subinterval [a,b) \subseteq [0,1), the proportion of terms \{n\alpha\} falling in [a,b) up to N approaches b-a as N \to \infty. Weyl generalized this to polynomial sequences. Consider a real polynomial P(t) = a_d t^d + \cdots + a_1 t + a_0 of degree d \geq 1. The sequence \{P(n)\}_{n=1}^\infty is equidistributed modulo 1 if and only if at least one coefficient a_j with $1 \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: \mathbb{Z} \to \mathbb{R}^m / \mathbb{Z}^m, where P(n) = (P_1(n), \dots, P_m(n)) and each P_i is a real polynomial. The sequence \{P(n)\}_{n=1}^\infty is equidistributed in the unit cube [0,1)^m if the leading coefficients of the P_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)\} modulo one to verifying that the exponential sums vanish in the limit: for every integer m \neq 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 N.^[9]^[11]^[5] In the linear case where P(k) = \alpha k and \alpha is irrational, the sum is a geometric series:

\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)|}.

Since m\alpha is not an integer, \sin(\pi m \alpha) \neq 0 and is bounded away from zero for fixed m, yielding a bound of O(1). Thus, the normalized sum is at most $1 / (N |\sin(\pi m \alpha)|), which tends to zero as N \to \infty.^[9]^[11]^[5] For polynomials of higher degree d \geq 2, Weyl's differencing method or the van der Corput inequality is employed to bound the sums by iteratively estimating differences, achieving decay of order O(N^{1 - \delta}) for some \delta > 0. Specifically, for the quadratic case P(k) = \alpha k^2 + \beta k + \gamma with \alpha irrational, square the sum S = \sum_{k=1}^N e^{2\pi i m P(k)} and apply the Cauchy-Schwarz inequality:

|S|^2 = \sum_{h=1}^N \sum_{k=1}^{N-h} e^{2\pi i m (P(k+h) - P(k))}.

The inner sum over k then involves a phase P(k+h) - P(k) = 2\alpha h k + (\alpha h^2 + \beta h), which is linear in k; applying the linear case bound yields |S|^2 \ll N^{3/2}, so |S| \ll N^{3/4}, and the normalized sum tends to zero.^[9]^[11]^[5] This squaring technique generalizes via iterated differencing for degree d: repeated applications of the difference operator \Delta_h P(k) = P(k+h) - P(k) reduce the polynomial degree by one each time, eventually yielding linear phases whose sums are controllable, ensuring sublinear growth |S| = O(N^{1 - 1/(2^{d-1})}) and thus equidistribution when the leading coefficient is irrational. The van der Corput inequality provides a refined tool for these bounds, stating that for a suitable difference operator, the sum magnitude is controlled by averages over shifts h, facilitating the inductive step.^[9]^[11]^[5]

Examples and applications

Basic examples

A fundamental example of an equidistributed sequence is the fractional parts of multiples of an irrational number, specifically \{n \sqrt{2}\} for n = 1, 2, 3, \dots, where \{\cdot\} denotes the fractional part. This sequence is equidistributed modulo 1, meaning that as N increases, the points fill the interval [0,1) uniformly, with the proportion of terms falling into any subinterval [a,b) \subset [0,1) approaching b - a.^[12] In visualizations such as histograms of the first N terms, the bars approximate a uniform density across [0,1), becoming smoother and closer to the constant height 1 as N \to \infty, illustrating the even spreading without gaps or clusters.^[13] Another illustrative case is the quadratic sequence \{n^2 \alpha\} where \alpha is irrational. Despite the accelerating quadratic growth, which introduces curvature in the underlying progression, the fractional parts remain equidistributed modulo 1, uniformly populating [0,1) in the limit.^[14] Histograms of partial sequences here also converge to uniform density, demonstrating how the theorem accommodates polynomial distortions while preserving overall uniformity.^[13] In contrast, consider the sequence \{n \alpha\} where \alpha is rational, say \alpha = p/q in lowest terms with integers p, q > 0. This sequence takes only q distinct values modulo 1, repeatedly cycling through them, leading to clustering at those fixed points rather than uniform distribution across [0,1).^[10] Histograms of such sequences show discrete spikes at the rational points, with empty regions elsewhere, clearly failing equidistribution. Weyl's criterion provides the general framework for verifying these behaviors across polynomial sequences.

Applications in number theory

The equidistribution of the sequence \{n \alpha\} for irrational \alpha, where \{\cdot\} denotes the fractional part, plays a crucial role in the distribution of lattice points near the line y = \alpha x in the plane. Specifically, it implies that the lattice points (n, \lfloor n \alpha \rfloor) are uniformly distributed along this line modulo the torus, providing insights into the error term in the Gauss circle problem and related lattice point discrepancies. This uniformity extends to the "hitting" of Farey fractions, where the sequence \{n \alpha\} intersects intervals defined by adjacent Farey fractions of order Q in a manner proportional to their lengths, ensuring an even coverage of rational approximations up to denominator Q.^[15] Furthermore, this equidistribution connects to continued fraction expansions, as the positions of \{n \alpha\} relative to Farey arcs determine the semi-convergents, leading to a uniform distribution of the approximants in the Farey diagram.^[16] In analytic number theory, the equidistribution theorem facilitates estimates of exponential sums over primes via Weyl sums. Vinogradov's theorem establishes that \{\alpha p\} is equidistributed modulo 1 for irrational \alpha, where p ranges over primes, relying on bounds for sums like \sum_p e^{2\pi i P(p)} with polynomial P. These estimates, derived from the equidistribution criterion, have applications to variants of the Goldbach conjecture, such as the ternary Goldbach problem, where controlling the distribution of primes modulo 1 helps resolve additive questions about sums of three primes.^[17] The nontrivial bounds on such Weyl sums over primes, often O(x (\log x)^{-c}) for some c > 0, underscore the theorem's role in bridging uniform distribution and sieve methods.^[18] The equidistribution theorem provides quantitative bounds in Diophantine approximation through the concept of discrepancy, which measures the deviation from uniformity. For the sequence \{n \alpha\}, the discrepancy D_N satisfies D_N = o(1) as N \to \infty, and more precise estimates link it to the quality of rational approximations to \alpha; specifically, small discrepancies imply that \alpha cannot be too well approximated by rationals beyond Dirichlet's theorem. This connection yields bounds like \|q \alpha\| \gg 1/(q D_q), where the equidistribution ensures that for almost all irrationals, the approximation exponent remains bounded, refining classical results on how well irrationals can be approximated.^[16] Such discrepancy-based approaches also inform the metric theory of approximation, distinguishing typical from exceptional irrationals.^[19] In the metric theory of Diophantine approximation, the equidistribution theorem implies that for almost all real \alpha (in the Lebesgue measure sense), the sequence \{n^2 \alpha\} is equidistributed modulo 1, as the quadratic coefficient \alpha is irrational with probability 1. Weyl's generalization to polynomials confirms this for any quadratic with irrational leading coefficient, and the exceptional set—where \alpha is rational—has measure zero. This has measure-theoretic implications for badly approximable numbers, which form a set of measure zero but full Hausdorff dimension 1; despite their poor approximability (\|q \alpha\| > c/q for some c > 0), they still exhibit equidistribution for \{n^2 \alpha\}, highlighting the robustness of the theorem across the irrationals while underscoring the null measure of these "badly" behaved points in the broader metric landscape.^[19]^[20]

Historical development

Early origins

The roots of the equidistribution theorem lie in the late 19th-century studies of dynamical systems, where concepts of uniform distribution emerged in the context of recurrence and trajectory behavior. In 1890, Henri Poincaré's work on the three-body problem and celestial mechanics explored the long-term behavior of orbits in phase space, introducing ideas that anticipated uniform distribution as a property of invariant measures in dynamics. His investigations emphasized how trajectories in conservative systems tend to fill space uniformly over time, setting the stage for later probabilistic interpretations.^[21] A key precursor came in 1884 with Leopold Kronecker's theorem on the density of the sequence of fractional parts {n \alpha} for irrational \alpha in the unit interval [0,1), providing early evidence for more refined distribution properties like equidistribution. This result built on approximation techniques and highlighted the irregular yet pervasive spreading of such sequences, influencing subsequent number-theoretic inquiries. Émile Borel advanced these ideas in 1909 with his theorem on normal numbers, demonstrating that almost all real numbers in [0,1] are normal in every integer base b \geq 2, meaning their base-b expansions contain each digit with asymptotic frequency 1/b. This normality condition implies equidistribution of the digit sequences modulo b for almost all such numbers, extending distributional uniformity to probabilistic settings and underscoring that equidistribution holds for a set of full Lebesgue measure.^[22] In the early 20th century, these developments connected to the emerging foundations of ergodic theory, notably through George David Birkhoff's 1931 ergodic theorem, which formalized pointwise convergence of time averages to space averages under measure-preserving transformations. Although postdating initial precursors, Birkhoff's result provided a general dynamical framework interpreting equidistribution as a consequence of ergodicity in systems like irrational rotations on the torus.^[23]

Key advancements

The basic equidistribution theorem for the linear sequence {n \alpha} modulo 1, with irrational \alpha, was proved in 1909 by Hermann Weyl and independently in 1910 by Wacław Sierpiński and Piers Bohl. In 1916, Weyl introduced the equidistribution theorem for polynomial sequences modulo one, establishing that if a polynomial with at least one irrational coefficient (other than the constant term) evaluates at consecutive integers, the resulting sequence is equidistributed in the unit interval.^[9] This breakthrough relied on estimates of exponential sums, providing a criterion based on the vanishing of Fourier coefficients for non-constant frequencies.^[9] During the 1920s, J. G. van der Corput advanced Weyl's framework through differencing techniques that improved bounds on exponential sums, enabling stronger estimates for the equidistribution of sequences derived from polynomials and other arithmetic progressions.^[24] These methods, including the van der Corput difference theorem, iteratively apply differences to reduce the problem to lower-degree cases, facilitating proofs of equidistribution under milder irrationality conditions.^[24] Post-World War II developments culminated in the 1974 monograph by Laurens Kuipers and Harald Niederreiter, which systematized the theory of uniform distribution and introduced discrepancy as a quantitative measure of equidistribution quality for sequences.^[5] Their work compiled and refined earlier results, emphasizing metric aspects and applications to multidimensional settings, thereby solidifying discrepancy theory as a cornerstone for assessing distribution irregularities.^[5] Recent advances as of 2024 have explored metric equidistribution for random polynomials, where the zeros or values distribute according to expected measures with high probability under random coefficient models.^[25] Concurrently, connections to quantum unique ergodicity in spectral geometry have linked equidistribution principles to the asymptotic behavior of eigenfunctions on manifolds, confirming uniform distribution in phase space for certain arithmetic families of Laplacians.^[26]

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