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Weyl’s equidistribution theorem for linear and quadratic polynomials Sayantan Khan July 2016 Given a polynomial p2R[X], one can look at the orbit of the polynomial restricted to the interval [0;1] The orbit is the set fp(n) mod 1 jn2Ng If p(x) is the polynomial ax, where ais an irrational number, then it’s not very di cult

Here is an elementary proof for the equidistribution of orbits of irrational rotations. Let $\alpha$ be any irrational number and fix $0\le a<b\le 1$. We need to show that \begin{equation}\tag{1}\label{eq:equidistribution} \frac{\# \left\{0\le j < n\,:\,\{j\alpha\} \in[a,b]\right\}}{n}\xrightarrow[n\to\infty]{}b-a. \end{equation}

The equidistribution theorem: The sequence of all multiples of an irrational α, 0, α, 2α, 3α, 4α, ... is equidistributed modulo 1. More generally, if p is a polynomial with at least one coefficient other than the constant term irrational then the sequence p(n) is uniformly distributed modulo 1.

Kronecker’s approximation theorem Weyl equidistribution 1. Dirichlet’s pigeon-hole principle, approximation theorem The pigeon-hole principle[1] formulated by Dirichlet by 1834, observes that when N+1 things are partitioned into Ndisjoint subsets, there …

The Theorem If g is irrational then for a,b 2[0,1] we have 1 n cardf1 r n : a hrgi bg!b a asn !¥ (1) Note that for rational g equidistribution will not occur. The fractional parts of integer multiples of a rational number will always fall on multiples

One reference is Theorem 6 of Chapter XV (Density of Primes and Tauberian Theorem) in. S. Lang: Algebraic Number Theory (Addison-Wesley, 1970). This is probably more general than Hecke's result, but the case of "equidistribution of ideals and primes in sectors" of the Gaussian numbers is singled out as Example 2 on page 318.

An Elementary Proof for the Equidistribution Theorem. The Mathematical Intelligencer September 2015, Volume 37, Issue 3, pp 1–2. Unfortunately the article is behind a paywall. The proof makes use of the following elementary criterium for equidistribution. As usual, denotes the fractional part of a real number. LEMMA 1.

We prove this analogue as a Corollary of a Theorem that guarantees equidistribution of lattice evaluations mod 1 for all functions which ...

[3.1] One-dimensional equidistribution With various formulations of integral, the stronger notion of equidistribution is equivalent to lim N!+1 1 N XN ‘=1 f( ‘) = Z 1 0 f(x) dx (for every Z-periodic f2C1(R)) [3.1.1] Theorem: (Weyl) A sequence f ‘gis equidistributed modulo Z if and only if lim N 1 N XN ‘=1 e2ˇin ‘ = 0 (for every n6= 0)

as Weyl's equidistribution theorem. Weyl worked in di- verse spheres of mathematics, among them, continuous groups and matrix representations. It was during his re- search into representation theory that Weyl discovered his theorem on equidistribution. Subsequently a vast

theory: Weyl’s Equidistribution Theorem. Terminology and Motivation Equidistribution A sequence of real numbers is said to be equidistributed if the quantity of terms which fall within an interval is proportional only to the length of the interval. Weyl’s Equidistribution theorem defines a class of such sequences:

Weyl's equidistribution theorem ... sequence {ξn} is said to be uniformly distributed (or equidistributed) on [0 ... theorem can be formulated as follows:.

Let us get back to the problem of equidistribution of If a C (0, 1), let us now evaluate the number of integers n such that e [0, For any n, let d = [v/n-I, the greatest integer less than or equal to v ~. Now, 0 _< (V/~/ _< a implies that d _< x/~<_ d+a. So, d 2 <_ n _< (d+a) 2 = d2+2da+a 2. For

A sequence (a1, a2, a3, ...) of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by (an) or by an − ⌊an⌋, is equidistributed in the interval [0, 1]. • The equidistribution theorem: The sequence of all multiples of an irrational α,0, α, 2α, 3α, 4α, ... is equidistributed modulo 1.

as Weyl's equidistribution theorem. Weyl worked in di- ... his theorem on equidistribution. ... quence ((u,)) of its fractional parts is equidistributed in.

In mathematics, the equidistribution theorem is the statement that the sequence. a, 2 a, 3 a, ... mod 1. is uniformly distributed on the circle. R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } , when a is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure.

The equidistribution theorem asserts that if {\alpha \in {\bf R}/{\bf Z} is an irrational phase, then the sequence {(n\alpha)_{n=1}^\infty} ...

16.07.2021 · Theorem 1.4 now gives the equidistribution of A q for most moduli q for which f ≡ 0 (mod q) has a root, and when the prime factors of q are constrained to the set P. For example, if m ⩾ 1 is a fixed integer, this applies to P being the set of primes of the form x 2 + m y 2.

As a generalization of Weyl's criterion, {uk}k⩾1 is equidistributed in [0,1]2 if and only if ∀ℓ∈Z2∖{0,0},limn→+∞1nn−1∑k=0e2iπℓ⋅uk=0.

In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1is 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 .

I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence α,2α,3α ...

Equidistribution in homogeneous spaces 3 x6. We conclude with a discussion of the quantitative aspects of the density and equidistribution results presented in the previous sections regarding orbits of group actions on homogeneous spaces. 2. Actions of …

The stronger version of Weyl’s equidistribution theorem states that the sequence q k= k2 + k+ mod 1 is equidistributed if is irrational. This will involve showing that the sequence S n= Xn k=1 exp(2ˇiaq k) (13) is o(n) for all integers n, but since is irrational, it su ces to show it for all irrational

Weyl's Equidistribution theorem defines a class of such sequences: the fractional parts of integer multiples of irrational numbers.

Equidistribution theorem ; In mathematics ; is uniformly distributed ; While this theorem was proved in 1909 and 1910 separately by Hermann Weyl ; In 1916, Weyl ...

Weyl's equidistribution theorem for linear and quadratic polynomials. Sayantan Khan. July 2016. Given a polynomial p ∈ R[X], one can look at the orbit of ...