6. Uniform distribution mod 1

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Winfred Underwood
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1 6. Uiform distributio mod Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which is less tha or eual to x ) ad its fractioal part {x } = x [x ]. Clearly, {x } < 1. The study of x mod 1 is the study of the seuece {x } i [, 1). Defiitio. We say that the seuece x is uiformly distributed mod 1 if for every a, b with a < b < 1, we have that 1 card{j j 1, {x j} [a, b]} b a, as. (The coditio is sayig that the proportio of the seuece {x } lyig i [a, b] coverges to b a, the legth of the iterval.) Remark. We ca replace [a, b] by [a, b), (a, b] or (a, b) with the same result. Exercise 6.1 Show that if x is uiformly distributed mod 1 the {x } is dese i [, 1). The followig result gives a ecessary ad sufficiet coditio for x to be uiformly distributed mod 1. Theorem 6.1 (Weyl s Criterio) The followig are euivalet: (i) the seuece x is uiformly distributed mod 1; (ii) for each l Z \ {}, we have as. e 2πilx j 1

2 6.2 The seuece x = α The behaviour of the seuece x = α depeds o whether α is ratioal or irratioal. If α Q, it is easy to see that {α} ca take o oly fiitely may values i [, 1): if α = p/ (p Z, N, hcf(p, ) = 1) the {α} takes the values { ( 1)p, { } p, { } 2p,..., I particular, {α} is ot uiformly distributed mod 1. If α R \ Q the the situatio is completely differet. We shall apply Weyl s Criterio. For l Z \ {}, e 2πilα 1, so we have Hece e 2πiljα = 1 e 2πilα 1 e 2πilα 1. e 2πiljα }. 1 2 e 2πilα, as. 1 Hece α is uiformly distributed mod 1. Remarks. 1. More geerally, we could cosider the seuece x = α + β. It is easy to see by modifyig the above argumets that x is uiformly distributed mod 1 if ad oly if α is irratioal. 2. Fix α > 1 ad cosider the seuece x = α x, for some x (, 1). The it is possible to show that for almost every x, the seuece x is uiformly distributed mod 1. We will prove this later i the course, at least i the cases whe α = 2, 3, 4, Suppose i the above remark we fix x = 1 ad cosider the seuece x = α. The oe ca show that x is uiformly distributed mod 1 for almost all α > 1. However, ot a sigle example of such a α is kow! I fact, ot a sigle example of a α for which α mod 1 is dese is kow. (Eve (3/2) mod 1 is ot kow to be dese.) Exercise 6.2 Calculate the freuecy with which 2 has r (r = 1,..., 9) as the leadig digit of its base 1 represetatio. (You may assume that log 1 2 is irratioal.) (Hit: first show that 2 has leadig digit r if ad oly if for some l Z +.) r 1 l 2 < (r + 1)1 l 2

3 Exercise 6.3 Calculate the freuecy with which 2 has r (r =, 1,..., 9) as the secod digit of its base 1 represetatio. 6.3 Proof of Weyl s criterio Remark. I the followig proof of Weyl s criterio, we assume some familiarity with properties of the Riema itegral. This was discussed i, for example, MT2222 Real Aalysis. Proof. Sice e 2πix j = e 2πi{xj}, we may suppose, without loss of geerality, that x j = {x j }. (i) (ii): Suppose that x j is uiformly distributed mod 1. If χ [a,b] is the characteristic fuctio of the iterval [a, b], the we may rewrite the defiitio of uiform distributio i the form χ [a,b] (x j ) From this we deduce that f(x j ) χ [a,b] (x) dx, as. f(x) dx, as, wheever f is a step fuctio, i.e., a liear combiatio of characteristic fuctios of itervals. Now let g be a cotiuous fuctio o [, 1] (with g() = g(1)). The, give ε >, we ca fid a step fuctio f with g f ε. We have the estimate g(x j ) g(x) dx (g(x j ) f(x j )) + f(x j ) f(x) dx + f(x) dx g(x) dx 2ε + f(x j ) f(x) dx. i= Sice the last term coverges to zero, we thus obtai lim sup g(x j ) g(x) dx 2ε. 3

4 Sice ε > is arbitrary, this gives us that g(x j ) g(x) dx, as, ad this holds, i particular, for g(x) = e 2πilx. If l the e 2πilx dx =, so the first implicatio is proved. (ii) (i): Suppose ow that Weyl s Criterio holds. The g(x j ) g(x) dx, as, wheever g(x) = m k=1 α ke 2πil kx is a trigoometric polyomial. Let f be ay cotiuous fuctio o [, 1] with f() = f(1). Give ε > we ca fid a trigoometric polyomial g such that f g ε. As i the first part of the proof, we ca coclude that f(x j ) f(x) dx, as. Now cosider the iterval [a, b] [, 1). Give ε >, we ca fid cotiuous fuctios f 1, f 2 (with f 1 () = f 1 (1), f 2 () = f 2 (1)) such that f 1 χ [a,b] f 2 ad f 2 (x) f 1 (x) dx ε. We the have that ad lim if lim sup χ [a,b] (x j ) lim if χ [a,b] (x j ) lim sup f 1 (x j ) = f 2 (x) dx ε f 2 (x j ) = f 1 (x) dx + ε 4 f 1 (x) dx χ [a,b] (x) dx ε f 2 (x) dx χ [a,b] (x) dx + ε.

5 Sice ε > is arbitrary, we have show that lim χ [a,b] (x j ) = χ [a,b] (x) dx = b a, so that x i is uiformly distributed mod 1. 5

f(x)g(x) dx is an inner product on D.

f(x)g(x) dx is an inner product on D. Ark9: Exercises for MAT2400 Fourier series The exercises o this sheet cover the sectios 4.9 to 4.13. They are iteded for the groups o Thursday, April 12 ad Friday, March 30 ad April 13. NB: No group o