2m 2 = + O(1) = m + O(1);

Transcription

1 UNIFORM DISTRIBUTION Andew Ganville Univesité de Montéal Zeev Rudnick Tel-Aviv Univesity 1. Unifom distibution mod one At pimay school the fist autho was taught to estimate the aea of a (convex) body by dawing it on a piece of gaph pape, and then counting the numbe of (unit) squaes inside. Thee is obviously a little ambiguity in deciding how to count the squaes which staddle the bounday. Whateve the potocol, if the bounday is moe-o-less smooth then the numbe of squaes in question is popotional to the peimete of the body, which will be small compaed to the aea (if the body is big enough). At seconday school the fist autho leant that thee ae othe methods to detemine aeas, sometimes moe pecise. As an undegaduate he leaned that counting lattice points is often a difficult question (and that counting unit squaes is equivalent to counting the lattice points in thei bottom left-hand cone). Then, as a gaduate student, he leant that the pimay school method could be tuned aound to povide a good tool fo estimating the numbe of lattice points inside a convex body! In the specific case of a ight-angled tiangle we fix the slope α of the hypotenuse and ask fo the numbe of lattice points A α (N) := #{(x, y) Z : x, y 0 and y + αx N}. Fo fixed α the pimay school method yields A α (N) = N α + O α(n). (1) Can we impove on the eo tem O α (N)? Fo intege m we have ) (A 1 (m) m A 1(m 1 m ) (m 1 m ) = A 1 (m) A 1 (m 1) ( ) ( ) m + 1 m m = + O(1) = m + O(1); c 006 Spinge. Pinted in the Nethelands.

2 A. GRANVILLE AND Z. RUDNICK thus we cannot eplace the O 1 (N) tem in (1) by o 1 (N). Moeove a simila agument woks wheneve α Q. It is unclea whethe (1) can be impoved when α Q so we now examine this case in moe detail: Fo each intege x 0 the numbe of integes y 0 fo which y + αx N is simply max{0, 1 + [N αx]}, whee [t] denotes the lagest intege t. Wheneve x N/α we can wite 1 + [N αx] = 1 + N αx {N αx}, whee {t} = t [t]. Theefoe [N/α] A α (N) = (1 + N αx {N αx}) x=0 = N α + 1 ( N + N ) [N/α] + O(1) α x=0 ({N αx} 1 ). () The fist tem is indeed the aea of ou tiangle. The second two tems account fo half the length of the peimete of ou tiangle. So, to able to pove that A α (N) = Aea + Peimete + o α (N), we need to establish that the mean value of {N αx} is 1 when α is iational, as one might guess. Actually we will pove something much stonge. We will pove that these values, in fact any set of values {αn + β : 1 n N} with α iational, ae unifomly distibuted mod one, so that thei aveage is 1 : DEFINITION.. A sequence of eal numbes a 1, a,... is unifomly distibuted mod one if, fo all 0 b < c 1 we have #{n N : b < {a n } c} (c b)n as N. Note that the values a n = np/q + β, 1 n N (hee α = p/q Q) ae evidently not unifomly distibuted mod one. Diichlet poved that fo any intege M 1 thee exists intege m, 1 m M such that mα < 1/M (whee t denotes the distance fom t to the neaest intege). To pove this note that thee ae M + 1 numbes {0 α}, {1 α},..., {M α} so, by the pigeonhole pinciple two, say {i α} and { j α} with 0 i < j M, must belong to the same inteval [k/m, (k + 1)/M) and so the esult follows with m = j i. Fo α Q we have δ := mα > 0. We will show that fo each i, 1 i m the set of values {αn + β : 1 n N, n i (mod m)} is well-distibuted mod one, and so the union of these sets is. This set of values is { j(mα)+(iα+β)} : 1 j J i } whee J i = N/m + O(1). We can ewite this as {δ j + γ (mod 1) : 1 j J}

3 UNIFORM DISTRIBUTION 3 whee γ iα + β (mod 1) if δ = {mα}, and γ iα + β δ(j + 1) (mod 1) if 1 δ = {mα}, by eplacing j with J + 1 j. Now, fo 0 γ < 1 and K = [δj + γ] #{ j J : {δ j + γ} [b, c)} = K #{ j J : δ j + γ [k + b, k + c)} k=0 = ( K + O(1) ) ( ) c b + O(1) δ ( ) c b = (c b)j + O + δj + 1. δ So fix ɛ > 0 and let M > 1/ɛ so that δ < 1/M < ɛ. We have just shown that ( m ) #{n N : {αn + β [b, c)}} = (c b)n + O δ + δn. Selecting N > m/δ this is ( c b + O(ɛ) ) N. Letting ɛ 0 we deduce that the sequence {αn + β : n 1} is unifomly distibuted mod one. The above agument woks fo linea polynomials in α but it is had to see how it can be modified fo moe geneal sequences. Howeve to detemine whethe a sequence of eal numbes is unifomly distibuted we have the following extaodinay, and widely applicable, citeion: WEYL S CRITERION. (Weyl, 1914) A sequence of eal numbes a 1, a,... is unifomly distibuted mod one if and only if fo evey intege b 0 we have e(ba n ) = o b(n) as N. (3) In othe wods lim sup N 1 N e(ba n ) = 0. (Hee, and thoughout, e(t) := e iπt.) In paticula if a n = αn + β then e(ba n ) = e(bβ) e(bαn) = e ( b(α + β) ) e(bαn) 1 e(bα) 1, the sum of a geometic pogession, povided bα is not an intege, so that e(ba n ) e(bα) 1 1 bα b 1 = o b (N), (4) as e(t) 1 t. Since bα is neve an intege when α Q we deduce, fom Weyl s citeion, that the sequence {αn + β : n 1} with α iational, is unifomly distibuted mod one.

4 4 A. GRANVILLE AND Z. RUDNICK REMARK. We immediately deduce fom Weyl s citeion that if a 1, a,... is unifomly distibuted mod one then so is ka 1, ka,... fo any non-zeo intege k. Actually this can be deduced fom the definition of unifom distibution mod one. Poof. We ecall that sin t t so that e(t) 1 π t. We begin by assuming that a 1, a,... is unifomly distibuted mod one. Fix intege b and then fix intege M > b. Since the sequence is unifomly distibuted mod one we know that fo each m, 0 m M 1, thee ae N/M + o(n) values of n N with m/m a n < (m + 1)/M; moeove, fo such n, we have e(ba n ) e(bm/m) π b/m. Theefoe e(ba n ) = M 1 m=0 ( N M + o(n) ) ( e ( ) ( )) bm 1 ( N ) + O b = O b + o(mn). M M M Now letting M get inceasingly lage we deduce that ou sum is indeed o b (N). On the othe hand, assume that (1) holds and define the chaacteistic function χ (b,c] by χ (b,c] (t) = 1 if {t} (b, c], and = 0 othewise. A well-known esult fom Fouie analysis tells us that one can appoximate any easonable function abitaily well using polynomials. That is, fo any ɛ > 0 thee exists intege d and coefficients c j, d j d, such that ( ) χ(t) f e(t) ɛ fo all t [0, 1) whee f (x) = j: j d c j x j. Theefoe ( ( #{n N : b < {a n } c} = χ (b,c] (a n ) = f e(an ) ) + O(ɛ) ) = j: j d c j e( ja n ) + O(Nɛ) = c 0 N + o(n) + O(Nɛ) by (4). Now c b = 1 0 χ (b,c] (t)dt = j: j d 1 c j e( jt) dt + O(ɛ) = c 0 + O(ɛ) 0 and so, by combining the last two equations and letting ɛ 0, we have shown that the sequence is unifomly distibuted mod one. One can deduce that a 1, a,... is unifomly distibuted mod one if and only if, fo evey continuous function f : [0, 1) R, we have 1 lim f ({a n }) = N N 1 0 f (x) dx. To pove this note that the functions e(bx), b Z 0 fom an appopiate (Fouie) basis fo the continuous functions on [0, 1).

5 UNIFORM DISTRIBUTION 5 An explicit vesion of Weyl s esult, which is useful fo many applications, was given by Edős and Tuán (Edős and Tuán, 1948): Fo any sequence of eal numbes, and any 0 b < c 1 we have 1 N #{n N : b < {a n} c} (c b) 6 m π m b=1 1 b 1 e(ba n ) N. Thee is a nice application of Weyl s theoem in the theoy of elliptic cuves: Let E be an elliptic cuve defined ove Q and suppose that E has infinitely many ational points. Poincaé showed that the ational points fom an additive goup, and Modell poved Poincaé s conjectue that this goup has finite ank; in othe wods E(Q) is an additive goup of the fom Z T whee the tosion subgoup T (that is, the subgoup of points of finite ode) and ae finite. Let us suppose that P 1,..., P fom a basis fo the Z pat of E(Q): Fo any given ac A on E(R) we can ask what popotion of the points {n 1 P 1 + n P n P + t : 0 n 1,..., n N 1, t T} lie on A, as N? The connection with ou wok above lies in the Weiestass paameteization of E: Thee exists an isomophism : C/(Z + Zi) E; that is (v + w) = (v) + (w) fo all v, w C. So select z 1,..., z C such that (z j ) = P j and τ such that (τ) = t. The above question then becomes to detemine the popotion of the points {n 1 z 1 + n z + + n z + τ (mod Z + Zi) : 0 n 1,..., n N 1, τ 1 (T)} that lie on 1 (A), a two-dimensional unifom distibution question. Like this the popotion can be shown to be / (x,y) A dx y (x,y) E(R) (Fo moe backgound on elliptic cuves see (Silveman and Tate, 199)). Fo given v = (a 1,..., a k ) R k define v (mod 1) to be the vecto (a 1 (mod 1),..., a k (mod 1)). We say that the sequence of vectos v 1, v,... R k is unifomly distibuted mod one if fo any 0 b j < c j < 1 fo j = 1,,..., k, we have { k } k # n N : a n (mod 1) [b j, c j ) (c j b j ) N as N. j=1 WEYL S CRITERION IN K DIMENSIONS. A sequence of vectos v 1, v,... R k is unifomly distibuted mod one if and only if fo evey b Z k, b 0 we have e(b.v n ) = o b(n) as N. (5) j=1 dx y.

6 6 A. GRANVILLE AND Z. RUDNICK We can deduce Konecke s famous esult that if 1, α 1, α,..., α k ae linealy independent ove Q then the vectos {(nα 1, nα,..., nα k ) : n 1} ae unifomly distibuted mod one. A final emak on {αn + β} n 1 : Let a n = αn + β (mod 1) fo all n 1. The tansfomation T α : x x + α gives T : a n a n+1. We want to define a measue µ on R/Z such that, fo any sensible set A we have µ(a) = µ(tα 1 A). In fact, when α Q, the only invaiant such measue, µ, is the Lebesgue measue, and thus the values a n ae distibuted accoding to this measue, that is they ae unifomly distibuted mod one. See Section.4 of Lindenstauss s pape in this volume (Lindenstauss, 006) fo moe details of this kind of egodic theoetic poof.. Factional Pats of αn We have seen, in the last section, that any sequence {αn + β : n 1}, with α iational, is unifomly distibuted mod one. One might ask about highe degee polynomials in n. Ou goal in this section is to pove the following celebated theoem of H. Weyl: THEOREM.1. Fo any iational eal numbe α, the sequence {αn : n 1} is unifomly distibuted mod one. Fo a steamlined poof, see the book (Kuipes and Niedeeite, 1974). Hee we will give an agument close to the oiginal: By Weyl s citeion, we need to show that fo fixed intege b 0, the Weyl sum N S β (N) = e(βn ) n=1 is o β (N), whee β = bα. Note that β is also iational. Weyl s idea was to squae the sum and notice that the esulting sum is essentially that of a polynomial one degee lowe, that is a linea polynomial. Indeed, S β (N) = e ( β(x y ) ) = N + R e ( β(y x ) ) x,y N Witing y = x+h, with h = 1,..., N 1, x = 1,..., N h we have y x = hx+h which is linea in x. Thus we find y>x N 1 N h S β (N) = N + R e(βh ) e(βh x) h=1 x=1

7 UNIFORM DISTRIBUTION 7 N 1 N h N + e(βh x) h=1 x=1 N 1 { } 1 N + min N,, (6) βh h=1 poceeding as in (4). We again use Diichlet s obsevation that thee exists q N with q(β) < 1/N. Let a be the intege neaest q(β); we may assume that (a, q) = 1. If h = H+ j, 1 j q then βh = βh +a j/q +O(1/N); so as j uns fom 1 to q the values βh (whee h = H + j) un though the values γ + i/q fo 0 i q 1, with eo no moe than O(1/N), whee γ 1/q. Thus, H+q h=h+1 { } 1 q/ min N, N + βh i=1 q i N + q log q. Patitioning the integes up to N 1 into at most N/q + 1 N/q intevals of length q o less, we thus deduce, fom (6), that 1 N S β(n) 1 q + log q N. (7) Now q = q N as N so (7) is o(1) and we ae done. To see that q N as N, suppose not so that q(β) < 1 N fo infinitely many integes N and thus q(β) = 0. But then β can be witten as a ational numbe with denominato q, contadicting hypothesis. This esult is widely applicable and this poof is easily modified to fit a given situation. Fo example see the poof of Lemma 3. in Heath-Bown s pape (Heath-Bown, 006) in this volume. A athe elegant egodic theoetic poof of Theoem.1 is given in Section 3 of Lindenstauss s pape in this volume (Lindenstauss, 006). Theoem.1 is a special case of THEOREM.. Let P(x) = a d x d +a d 1 x d 1 + +a 1 x+a 0 be a polynomial with at least one of the coefficients a 1,... a d iational. Then the sequence {P(n) : n 1} is unifomly distibuted modulo 1. This can be poved along the same lines as Theoem.1 (the special case of the polynomial P(x) = αx ) except that a single squaing opeation will now poduce a polynomial of one degee less, not a linea one. One then iteates this pocedue to get back to the case of linea polynomials, see, e.g., (Davenpot, 005) fo details.

8 8 A. GRANVILLE AND Z. RUDNICK One can deduce fom Weyl s citeion in n-dimension and Theoem. that the vectos {(nα, n α,..., n k α): n 1} ae unifomly distibuted mod one if α is not ational (see also Lindenstauss s aticle (Lindenstauss, 006) in this volume). 3. Unifom distibution mod N Fo a given set A define A(x) = 1 if x A, and A(x) = 0 othewise. Also define the Fouie tansfom of A to be Â(b) := A(n)e(bn) = e(bn). n n Witing A N = {a j : 1 j N} the Weyl citeion becomes that a 1, a,... is unifomly distibuted mod one if and only if  N (b) = o b (N) fo evey non-zeo intege b. When A is a subset of the esidues mod N we define ( ) bn ( ) bn Â(b) := A(n)e = e. N N Let A be a set of integes, and let (t) N denote the least non-negative esidue of t (mod N) (so that (t) N = N{t/N}). The idea of unifom distibution mod N is suely something like: Fo all 0 b < c 1 and all m 0 (mod N) we have n A n A #{a A : bn < (ma) N cn} (c b) A. (8) One can only make such a definition if A (since this is an asymptotic fomula) but we ae often inteested in smalle sets A, indeed that ae a subset of {1,,..., N}; so we will wok with something motivated by, but diffeent fom, (8). Let us see how fa we can go to poving the analogy to Weyl s citeion. Fix ɛ > 0: Define Eo(A; k) := max 0 x N m 0 (mod N) {a # A : x < (ma) N x + N } k A k Suppose that Eo(A; k) ɛ A /k fo some k > 1/ɛ We poceed much as in the poof of Weyl s citeion above: Subdivide ou inteval (0, N] into subintevals I j := ( jn/k, ( j + 1)N/k], so that if (ma) N I j then e(ma/n) = e( j/k) + O(1/k). Theefoe Â(m) = k 1 j=0 e(ma/n) = a A (ma) N I j k 1 e( j/k) j=0 k Eo(A; k) + A k ɛ A. a A (ma) N I j 1 + O( A /k).

9 UNIFORM DISTRIBUTION 9 In the othe diection ou poof is somewhat diffeent fom that fo Weyl s citeion: We begin by supposing that Â(b) ɛ A fo all b 0 (mod N). Fo J = [δn] a A 1 (ma) N J 1 = J j=1 a A 1 N ( e ( ma j )) = J N N A + 1 Â(m) N 0 J ( j ) e. N If uns though the non-zeo integes in ( N/, N/] then J j=1 e( j/n) N/. Thus the second tem hee is, fo R N/(ɛ A ) 0 Â(m) 0 R Â(m) + R< N/ Â(m) 1/ 1/ (log R) max Â(s) + Â(m) s 0 1/ R< (log R)ɛ A + ( A N/R) 1/ ɛ A povided ɛ 1/ log(n/ A ). One can thus fomulate an appopiate analogy to Weyl s citeion along the lines: The Fouie tansfoms of A ae all small if and only if A and all its dilates ae unifomly distibuted. (A dilate of A is the set {ma : a A} fo some m 0 (mod N).) This esult is cental to the spectacula ecent pogess in hamonic analysis by Gowes et. al, (see (Ganville et al., 006)). To give one example of how such a notion can be used, we ask whethe a given set A of esidues mod N contains a non-tivial 3-tem aithmetic pogession? In othe wods we wish to find solutions to a + b = c with a, b, c A whee a b. PROPOSITION 3.1. If A is a subset of the esidues (mod N) whee N is odd, fo which Â(m) < A /N 1 wheneve m 0 (mod N) then A contains nontivial 3-tem aithmetic pogessions. Poof. Since (1/N) e(t/n)=0 unless t is divisible by N, whence it equals 1, we have that the numbe of 3-tem aithmetic pogessions in A is 1 ( ) (a + b c) e = 1 Â() Â( ). N N N a,b,c A The = 0 tem gives A 3 /N. We egad the emaining tems as eo tems, and bound them by thei absolute values, giving a contibution (taking m (mod N)) 1 N Â() max Â(m) = A max Â(m). m 0 m 0 j=1

10 10 A. GRANVILLE AND Z. RUDNICK Thee ae A tivial 3-tem aithmetic pogessions (of the fom a, a, a) so we have established that A has non-tivial 3-tem aithmetic pogessions when yielding the esult. A 3 /N A max Â(m) > A, m 0 Let us apply Poposition 3.1 to the sets { A δ := n (mod N) : n N < δ } fo N pime with 0 < δ < 1. Fo J = [δn/] we have ( mn ) 1 (  δ (m) = e e ( j ) n ) N N N n J j J so that  δ (m) 1 ( j ) ( ) e mn n N N e J j J N n. Now n e(mn/n) = 0 if m 0, and = N if m = 0. If 0 then n e ( (mn n )/N ) is a Gauss sum and so has absolute value N. Moeove J j J e( j/n) N/ fo 1 N/. Inputting all this above we obtain  δ (m) N log N fo each m 0 (mod N) and #A δ =  δ (0) = δn + O( N log N). Now, fo fixed δ > 0 we have poved that each  δ (m) = o(δ N), and so Poposition 3.1 implies that A δ contains non-tivial 3-tem aithmetic pogessions. In fact the poof of Poposition 3.1 yields that A δ has δ 3 N 3-tem aithmetic pogessions a, a + d, a + d. The pevious esult is in fact a special case of Roth s (Roth, 1953) theoem, which states that fo any δ > 0 if N is sufficiently lage then any subset A of {1,..., N} with moe than δn elements contains a non-tivial 3-tem aithmetic pogession. His poof is a little too complicated to discuss in detail hee but we will outline the main ideas. If δ > 3 then A must contain thee consecutive integes, so the esult follows. Othewise we poceed by a fom of induction, showing that if thee exists A {1,..., N}, with #A δn, which contains no non-tivial 3-tem aithmetic pogession then thee exists A {1,..., N }, with #A δ N, which contains no non-tivial 3-tem aithmetic pogession, whee δ = (1 + cδ)δ and N = [N 1/3 ]. The induction then yields Roth s theoem fo any δ 1/ log log N. To pove the induction step we begin by inceasing N by a negligible amount so that it is pime, and then consideing A as a set of esidues mod N. By a slight modification of the poof of Poposition 3.1 one can show that if A does not contain a non-tivial 3-tem aithmetic pogession then A is not

11 UNIFORM DISTRIBUTION 11 unifomly distibuted mod N. By the definition of unifomly distibuted mod N, this implies that thee is some dilate of A, say ma (mod N) and some segment [bn, cn] which contains athe moe o athe less elements than expected; one can show that, in fact, thee must be some segments with athe moe, and some segments with athe less. Taking one of these segments with athe moe elements than expected, in fact containing 1+cδ times as many elements as expected, we can identify a segment of an aithmetic pogession (of length N ) within {1,..., N} which contains δ N elements of A, and fom this we constuct A (intege j A if and only if the jth tem of the aithmetic pogession is in A). 4. Nomal Numbes Ae thee any pattens in the digits of π? Science fiction wites (Sagan, 1985) would have us believe that secet messages ae encoded fa off in the tail of π but computational evidence so fa suggests the contay, that thee ae no pattens, indeed that evey sequence of digits appeas about as often as in a andom sequence. If the digits ae witten in base 10 then this question is equivalent to asking whethe the sequence {10 n π : n 1} is unifomly distibuted mod one? If so we say that π is nomal in base 10. In geneal we say that eal numbe α is nomal in base b if the sequence {b n α : n 1} is unifomly distibuted mod one; and that α is nomal, if it is nomal in base b fo evey intege b. In geneal vey little is known about nomality. A few specific numbes of vey special fom can be shown to be nomal to cetain bases. The one thing that we can show is that almost all numbes ae nomal, with a poof that fails to identify any such numbe! THEOREM 4.1. Almost all x [0, 1) ae nomal. (By almost all we mean that the set of such x has measue 1.) Theoem 4.1 follows fom: THEOREM 4.. Fo any inceasing sequence of integes a 1, a,..., the sequence {a n x : n 1} is unifomly distibuted mod one fo almost all x [0, 1). Deduction of Theoem 4.1. Taking a j = b j fo each j we see that almost all x [0, 1) ae nomal in base b. Theoem 4.1 follows since the exceptional set has measue 0 as it is a countable union of measue 0 sets. Poof of Theoem 4.. We begin by noting that 1 1 e(ba n x) N dx = 1 1 N e ( bx(a m a n ) ) dx = 1 N ; 0 m, 0

12 1 A. GRANVILLE AND Z. RUDNICK so that m 1 m e(ba n x) n m 1 dx = m = π 6. m 1 Theefoe (in a step that takes some thinking about) 1 m 1 m e(ba n x) < n m fo almost all x, and so lim m 1 m e(ba n x) n m = 0. Now if m N < (m + 1) then e(ba n x) = n m e(ba nx) + O(m) and the esult follows. Refeences Davenpot, H. (005) Analytic methods fo Diophantine equations and Diophantine inequalities, Cambidge, Cambidge Univesity Pess. Edős, P. and Tuán, P. (1948) On a poblem in the theoy of unifom distibution I, II, Indag. Math. 10, , Ganville, A., Nathanson, M., and Solymosi, J. (eds.) (006) Additive Combinatoics, a school and wokshop, Povidence, RI, Ame. Math. Soc., to appea. Heath-Bown, D. R. (006) Analytic methods fo the distibution of ational points on algebaic vaieties, in this volume. Kuipes, L. and Niedeeite, H. (1974) Unifom distibution of sequences, Pue and Applied Mathematics, New Yok London Sydney, Wiley-Intescience. Lindenstauss, E. (006) Thee examples of how to use measue classification in numbe theoy, in this volume. Roth, K. F. (1953) On cetain sets of integes, J. London Math. Soc. 8, Sagan, C. (1985) Contact: A Novel, New Yok, Simon and Schuste. Silveman, J. and Tate, J. (199) Intoduction to elliptic cuves, New Yok, Spinge-Velag. Weyl, H. (1914) Übe ein Poblem aus dem Gebeit de diophantischen Appoximationen, Nach. Ges. Wiss. Göttingen (math.-phys. Kl.) pp

13 INDEX 3-tem aithmetic pogession, 9 elliptic cuves, 5 Fouie tansfom, 8 lattice points in a ight-angled tiangle, 1 nomal numbes, 11 unifomly distibuted mod one,, 5 Weyl s citeion, 3