Number Theory: An Introduction to Mathematics

446 X A Character Study [46] A. Weil,L’integration dans les groupes topologiques et ses applications, 2nd ed., Hermann, Paris, 1 ...

XI Uniform Distribution and Ergodic Theory A trajectory of a system which is evolving with time may be said to be ‘recurrent’ if ...

448 XI Uniform Distribution and Ergodic Theory For any real numberξ,letξdenote again the greatest integer≤ξand let {ξ}=ξ−ξ d ...

1 Uniform Distribution 449 Theorem 1A real sequence(ξn)is uniformly distributed mod 1 if and only if, for every function f:I→Cwh ...

450 XI Uniform Distribution and Ergodic Theory A converse of Theorem 1 has been proved by de Bruijn and Post (1968): if a func- ...

1 Uniform Distribution 451 If f is a continuous function of period 1 then, by the Weierstrass approximation theorem, for anyε> ...

452 XI Uniform Distribution and Ergodic Theory These results can be immediately extended to higher dimensions. A sequence (xn)of ...

1 Uniform Distribution 453 It follows that each point of the sequence(nx)lies on some hyperplanem·y =h, whereh ∈ Z. Without loss ...

454 XI Uniform Distribution and Ergodic Theory Proof By takingζn=e(ξn)in Lemma 4 we obtain, for 1≤M≤N, N−^2 ∣ ∣ ∣ ∣ ∑N n= 1 e(ξn ...

1 Uniform Distribution 455 is a polynomial of degreer−1 with leading coefficientrmαr.Ifαris irrational, then rmαris also irratio ...

456 XI Uniform Distribution and Ergodic Theory Since q−^1 ∑q k= 1 e(nk/q)=1ifn≡0modq, =0ifn≡0modq, we can write S=(qN)−^1 ∑qN n ...

1 Uniform Distribution 457 It may be noted that the matrixAin Proposition 9 is necessarily non-singular. For if detA =0, there e ...

458 XI Uniform Distribution and Ergodic Theory By the mean value theorem, the hypotheses of Proposition 10 are certainly satisfi ...

2 Discrepancy 459 2 Discrepancy................................................ Thestar discrepancyof a finite set of pointsξ 1 ...

460 XI Uniform Distribution and Ergodic Theory D∗N= max 1 ≤k≤N max(|k/N−ξk|,|(k− 1 )/N−ξk|). The second expression forD∗Nfollows ...

2 Discrepancy 461 Proof Without loss of generality we may assumeξ 1 ≤···≤ξN. Writing ∫ I f(t)dt= ∑N n= 1 ∫n/N (n− 1 )/N f(t)dt, ...

462 XI Uniform Distribution and Ergodic Theory As an application of Proposition 15 we prove Proposition 16Ifξ 1 ,...,ξNare point ...

2 Discrepancy 463 Propositions 14 and 15 show that in a formula for numerical integration the nodes (ξn)should be chosen to have ...

464 XI Uniform Distribution and Ergodic Theory Thus ifC∗is the least upper bound for all admissible values ofCin Schmidt’s resul ...

3 Birkhoff’s Ergodic Theorem 465 actual state of motion, will, sooner or later, pass through every phase which is con- sistent w ...