Fourier series, Weyl equidistribution. 1. Dirichlet s pigeon-hole principle, approximation theorem

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1 (October 4, 25) Fourier series, Weyl equidistribution Paul Garrett garrett/ [This document is garrett/m/mfms/notes 25-6/8 Fourier-Weyl.pdf] Dirichlet s pigeon-hole principle, approximation theorem Kronecker s approximation theorem Weyl equidistribution. Dirichlet s pigeon-hole principle, approximation theorem The pigeon-hole principle [] formulated by Dirichlet by 834, observes that when + things are partitioned into disjoint subsets, there is at least one subset containing at least 2 things. The archetypical application is the following: [..] Theorem: (Dirichlet) For every real α and every integer, there are integers p, q with q such that qα p. Proof: For each m in the range m +, choose n = n m so that mα n m [, ) is the fractional part [2] of mα. The + choices of m produce + numbers mα n in [, ). Dividing the interval into subintervals of length, by the pigeon-hole principle some subinterval contains both mα n and m α n for some m < m +. That is, (mα n) (m α n ) = (m m )α (n n ) so q = m m and p = n n meet the requirement of the theorem. /// [..2] Remark: It is trivial to see that, given a real number r and given a positive integer n, there is integer m such that r m n < n. The theorem comes close assuring a smaller error of n, but then we no 2 longer have freedom to choose the denominator. 2. Kronecker s approximation theorem The following is an example of Kronecker s 884 generalization of Dirichlet s approximation results, in which Kronecker illustrates a different causal mechanism: [2..] Theorem: The collection of integer multiples nα of irrational real α is dense in T = R/Z. Proof: The assertion is equivalent to Zα + Z being dense in R. The topological closure Γ of Zα + Z in R is still a subgroup of R. The topologically-closed subgroups of R are classifiable: they are exactly {}, free Z-modules Zβ on a single generator β, and the whole R. Granting this classification for a moment, if Γ is not the whole, then Γ = Z β for some β, and certainly Zα + Z Zβ. Thus, Z Zβ, so = nβ for some integer n, and β = n. Similarly, α = mβ = m n for some n, so α is rational. [] Dirichlet used Schubfachprinzip, which is drawer principle. [2] The fractional part of a real number α, sometimes denoted {α} or α or α mod, in an elementary context is α = α α, where α is the greatest integer less than or equal α. More to the point, the fractional part is really the image of α in the quotient R/Z.

2 Paul Garrett: Fourier series, Weyl equidistribution (October 4, 25) ow we classify topologically-closed subgroups Γ {} of R. Since Γ {} and is closed under additive inverses, Γ contains positive elements. In the case that there is a least positive element γ o, we claim that Γ = Z γ o. Indeed, given another < γ Γ, by the archimedean property of the real numbers there is an integer n such that nγ o γ < (n + )γ o. In fact, nγ o = γ, or else < γ nγ o < γ o, contradicting the minimality of γ o. In the case that there is no least positive γ o Γ, let γ > γ 2 >... > be an infinite sequence of positive elements of Γ. The infimum γ o of this sequence is in Γ, since Γ is closed. Replacing γ j by γ j γ o, we can assume that γ j. Given real β, there is integer n such that nγ j β < (n + )γ j by archimedean-ness, so Zγ j contains elements within γ j of any real number. Since γ j, given ε > there is γ j < ε, so every real number is within ε > of Zγ j. Thus, the topologically closed subgroup Γ must be R. /// A similar argument would prove Kronecker s multi-dimensional version: [2..2] Theorem: (Kronecker) An n-tuple α = (α,..., α n ) of real numbers is linearly independent over Q if and only if the topological closure of the collection { α : =, 2, 3,...} of multiples of α is dense in the n-torus T n = R n /Z n. /// 3. Weyl equidistribution criterion The idea of a sequence of real numbers α, α 2,... being equidistributed modulo Z, that is, in R/Z, is a quantitative strengthening of a merely qualitative density assertion. More generally, the notion of equidistribution of a sequence of finite sets S, S 2,... can be quantified, specializing to the case of a sequence α, α 2,... of points by taking S n = {α,..., α n }. [3..] Remark: In fact, to allow repeats of points, we are really looking at sequences of multi-sets, meaning that each element has a multiplicity that can be greater than, as opposed to genuine sets, which collapse multiplicities. For example, {a, b} = {a, b, b} = {a, b, b, b} =.... One possible equidistribution requirement is that the number of α n mod Z with n in any subinterval [a, b] of [, ] is asymptotically b a as. It is simpler to weaken the requirement a little, in effect replacing the collection of characteristic functions of subintervals [a, b] with arbitrary continuous functions on R/Z, and, further to consider the collection of all smooth functions on R/Z. Say that a sequence of points α, α 2,... is equidistributed mod Z when + f(α n ) = n= f(x) dx (for every f C c (R/Z)) Say that a sequence of (multi-) sets S, S 2,... is equidistributed mod Z when + #S α S f(α) = f(x) dx (for every f C c (R/Z)) We want to allow possible repetition in the elements in the S, so these are not properly set, since sets do not allow repetition: {a, b} = {a, b, b} = {a, b, b, b, } =.... Thus, these S s are multi-sets, or lists. Still, we will use set-element notation, and write α S to refer to all elements of the list S. Let ψ n (x) = e 2πinx. [3..2] Theorem: (Weyl) A sequence of (multi-) sets {S } is equidistributed mod Z if #S α S ψ n (α) = (for every n ) 2

3 Paul Garrett: Fourier series, Weyl equidistribution (October 4, 25) In particular, for irrational real α, the sequence α, 2α, 3α,... is equidistributed modulo Z. [3..3] Remark: The other direction of implication in the theorem is trivial, since the exponentials are smooth functions. That is, Weyl s theorem says that it suffices to check equidistribution relative to a restricted list of smooth functions, the exponentials ψ n. [3..4] Remark: The sums ψ n (α) = #S #S α S α S e 2πinα are reasonably construed as the Fourier coefficients of the measure f α S f(α). Proof: Smooth Z-periodic f has Fourier expansion n f(n) ψ n converging absolutely and uniformly to f. f(α) = ( ) f(n)ψ n (α) #S #S α S α S n n = f() + ( ) f(n) ψ n (α) #S n α S For any cut-off b for the Fourier series, noting f() = f(x) dx, f(α) #S α S f(x) dx < n b n ( f(n) #S = #S ( ) f(n) ψ n (α) α S ) ψ n (α) f(n) ψ n (α) #S α S α S n f(n) ψ n (α) + #S f(n) α S Since the Fourier series of f converges absolutely, given ε > there is large-enough b so that n >b f(n) < ε. With that b, since #S α S ψ n (α) for each fixed n, and since there are only finitely-many n with < n b, for large-enough Thus, That is, #S α S f(α) < n b n >b f(n) ψ n (α) < ε #S α S f(α) #S f() < 2ε α S f(x) dx, and Weyl s criterion suffices for equidistribution. In the example of integer multiples of an irrational number, by summing a geometric series, with n, e 2πin lα = e 2πinα e 2πin(+)α e 2πinα l= The irrationality of α and n assure that the denominator does not vanish. Thus, e 2πin lα 2 e 2πinα l= (for each fixed n ) proving equidistribution of {lα}. /// 3

4 Paul Garrett: Fourier series, Weyl equidistribution (October 4, 25) [3..5] Remark: The proof only used absolute convergence pointwise of the Fourier series of f to f. Infinitedifferentiability of f assures this, but much less is needed. On the other hand, if we allow merely-continuous functions f, then we must invoke Weierstrass approximation, since the Fourier series of typical continuous functions do not converge to them pointwise. [3.] Higher-dimension analogues Fourier series of functions f on R n /Z n can be written f(ξ) e 2πiξ x where ξ x = ξ x + ξ 2 x ξ n x n, and f(ξ) = ξ Z n... e 2πiξ x f(x) dx The notion of equidistribution modulo Z n of a sequence {α l } in R n is the same: l f(α l ) =... f(x) dx (for all Z n -periodic smooth f on R n ) Assuming we know that Fourier series of a nice Z n -periodic function f on R n converges absolutely to f pointwise, the same argument proves [3..] Theorem: (Weyl) A sequence {α l } in R n is equidistributed modulo Z n if and only if l e 2πiξ α l = (for all ξ Z n ) For example, for real numbers β,..., β n, the sequence α l = l (β,..., β n ) is equidistributed modulo Z n if and only if, β,..., β n are linearly independent over Q. /// [3..2] Remark: Weyl s criterion for equidistribution can be applied to compact topological groups K of various sorts, in place of R n /Z n, especially compact Lie groups, because of the spectral decomposition of L 2 (K) analogous to Fourier series on R n /Z n. In general, Stone-Weierstrass approximation must be sued. [3..3] Remark: Weyl treated a more serious problem, that of equidistribution modulo Z of sequences α l = P (l) with polynomial P. Bibliography [Dirichlet 829] P. G. L. Dirichlet, Sur la convergence des series trigonomeétriques que servent à représenter une function arbitraire entre des ites données, J. für die reine und angew. Math. 4 (829), [Dirichlet 863] P. G. L. Dirichlet, Vorlesungen über Zahlentheorie, Vieweg, 863. [Erdös-Turán 948] P. Erdös, P. Turán, On a problem in the theory of uniform distribution, I, II, ederl. Akad. Wetensch. 5, 46-54, [Fourier 87] J. Fourier, Mémoire sur la propagation de la chaleur dans les corps solides, ouveau Bulletin des Sciences par la Soc. Phil. de Paris I no. 6, (88), 2-6; Oeuvres complètes, tome 2, G. Darboux, [Kronecker 884] L. Kronecker, äherungsweise ganzzahlige Auflösung linearer Gleichungen, Werke 3, Chelsea, 968, 47-9 (original 884). 4

5 Paul Garrett: Fourier series, Weyl equidistribution (October 4, 25) [Miller 23] J. Miller, Earliest known uses of some of the words of mathematics, uploaded Sept. 4, 23. [Weyl 96] H. Weyl, Über die Gleichverteilung von Zahlen modulo Eins, Math. Ann. 77 no. 3 (96), [Weyl 92] H. Weyl, Zur Abschätzung von ζ( + it), Math. Zeit. (92),