Euidistribution and Wyl s critrion by Brad Hannigan-Daly W introduc th ida of a sunc of numbrs bing uidistributd (mod ), and w stat and prov a thorm of Hrmann Wyl which charactrizs such suncs. W also discuss a fw intrsting rsults that follow from Wyl s thorm. Wyl s uidistribution critrion Dfinition. Lt u, u,... b a boundd sunc of ral numbrs. W say that this sunc is uidistributd or uniformly distributd (mod ) if, for vry subintrval (α, β) of [, ], w hav {{u },..., {u }} (α, β) β α. (For ach x R, {x} dnots its fractional part x x.) That is to say, th proportion of th {u j } that li in any givn subintrval is proportional to th lngth of that subintrval, and thus this sunc of fractional parts is vnly distributd in [, ] Obsrv that for such a sunc, it immdiatly follows that {u }, {u },... is dns in [, ]: for any subintrval (α, β) [, ], sinc {{u },..., {u }} (α, β) β α > thr xists som last intgr such that {{u },..., {u }} (α, β), whnc {u } (α, β). W not also that w may rplac instancs of (α, β) by any of [α, β) in this dfinition sinc th diffrnc btwn {{u },..., {u }} (α, β) and {{u },..., {u }} [α, β) is at most, which vanishs in th it. A similar rmark holds for rplacing such instancs by (α, β] or [α, β]. Wyl s critrion provids a charactrization of th suncs that ar uidistributd (mod ) which, among othr things, implis that ustions about uidistribution can b rducd to finding bounds on crtain xponntial sums. Thorm (Wyl s critrion). Lt u, u,... b a sunc of ral numbrs. Th following ar uivalnt: () u, u,... is uidistributd (mod ) () For ach nonzro intgr k, w hav (ku n ) n () For ach Rimann-intgrabl f : [, ] C, w hav f({u n }) n f(x)dx Bfor procding to th proof of this rsult, w offr som huristic justification for it. Suppos w ar givn a sunc u,..., and th fractional parts {u },... ar placd on th intrval [,], which is thn Th notion of uidistributivity of a sunc is mor gnrally dfind in crtain classs of locally compact groups, using th Haar masur, and rsults such as analogus to Wyl s critrion can b provd in this mor gnral stting. Th original rsults can b rcovrd by considring th compact group T R/Z. S, for xampl, [].
wrappd around th unit circl k tims for som nonzro intgr k. W would xpct that, if th sunc is uidistributd (mod ), th corrsponding points on th circl should b also b vnly distributd. Th condition () stats that uidistribution (mod ) is uivalnt to th first of ths points having a cntroid which approachs th cntr of th circl as bcoms larg, rgardlss of th choic of k, which w would xpct of a sunc of points that is vnly distributd on th circl. Th third condition can b intrprtd as saying that, givn a sunc of numbrs in [,), it is uidistributd in [,] if and only if th avrag valu of ach intgrabl function on [,] can b obtaind by avraging ovr only th points of that sunc, which is a plausibl consunc of uidistribution. Proof. () () Lt f : [, ] C b an intgrabl function. Assum without loss of gnrality that f is ral-valud, sinc othrwis w can just considr th ral and imaginary parts sparatly. Lt I [α, β) [, ]. oting that {{u },..., {u }} [α, β) [α,β) ({u n }) and that I β α, w conclud from () that () holds in th cas that f is th charactristic function of an opn subintrval I, and th sam rasoning shows that this also holds if I is a closd or half-opn subintrval. ow, if λ, λ ar ral numbrs and f, f ar functions for which () holds, w hav (λ f + λ f )({u n }) λ n n n f ({u n }) + λ λ f + λ (λ f + λ f ) f f ({u n }) and w conclud that () holds for all R linar combinations of charactristic functions of subintrvals, hnc for all stp functions on [, ]. ow, lt f : [, ] R b an arbitrary intgrabl function, and lt ε >. Thn thr xist stp functions f, f : [, ] R such that f f f pointwis and (f f ) < ε. As f f, w hav n hnc (f f ) (f f) + f ε < (f f ) f f ({u n }). n (f f ) < ε, It follows that, for larg nough, n f ({u n }) > f ε and thus n f({u n}) > f ε for larg. W similarly obtain n f({u n}) < f + ε for larg, hnc n f({u n}) f < ε for sufficintly larg, proving th uality in (). () () As abov, taking f [α,β) for ach [α, β) [, ] shows that () holds. () () Fix k Z\{}, and lt f(x) (kx). Sinc f(x + ) f(x) for all x, it follows that f({x}) f(x) and so
th lft-hand sid of th uation in () is ual to th lft-hand sid in (). But th right-hand sid in () is (kx)dx cos(πkx) + i sin(πkx)dx, for k, and hnc () follows. () () As bfor, w nd only concrn ourslvs with ral-valud intgrabl functions. W procd by showing that () holds for all continuous functions on [, ], thn that it holds for all stp functions on [, ]. This is sufficint to prov (), as shown in th proof that () (). Clarly () holds for th constant function, sinc in this cas n f({u n }) As in th proof that () (), w also s that () implis immdiatly that () holds for th ral and imaginary parts of functions f of th form f(x) (kx) with k a nonzro intgr, hnc for all functions cos(πkx) and sin(πkx). It follows that () holds for all R-linar combinations of such functions and th constant function. Hnc it holds for all trigonomtric polynomials of th form (x) a + (a cos πx + b sin πx) + + (a r cos πrx + b r sin πrx) for a j, b j R. Lt f b a continuous ral-valud function on [, ] and fix ε >. By th Ston-Wirstrass thorm, thr xists a trigonomtric polynomial such that f < ε. Taking f ε and f + ε, w hav f f f and (f f ) ε. As bfor, w conclud that () holds for this choic of f. ow, if g is any stp function on [, ], w can find continuous functions g, g on [, ] with g g g and (g g ) < ε. W again conclud that () holds for g, as dsird.. Applications On of th most wll-known corollaris to Wyl s critrion is th following rsult. Corollary. Lt θ b an irrational numbr. Thn th sunc (nθ) n is uidistributd (mod ). Proof. W show that this sunc satisfis th condition (). Lt k b a nonzro intgr. Sinc θ is irrational, kθ is not an intgr and so πikθ is nonzro. Thn for ach, w hav (knθ) n (kθ) (k( + )θ) (kθ) (kθ) and this tnds to zro as, as dsird. Wyl gnralizd th abov corollary to th following: Thorm. Lt p(n) b a polynomial with ral cofficints such that som cofficint, othr than th constant trm, is irrational. Thn (p(n)) n is uidistributd (mod ).
To prov this rsult, Wyl introducd a gnral procdur for finding uppr bounds on xponntial sums of th form S(t) n (tf(n)) for crtain intgr-valud functions f. This tchniu has com to b known as Wyl diffrncing. Givn α R, w will giv a bound on th sum n (n α) which will, using th scond part of Wyl s critrion, show that th sunc (n α) n is uidistributd (mod ) in th cas that α is irrational. W first ruir two lmmata. For x R, w dnot by x th distanc min({x}, {x}) from x to th narst intgr. Lmma. Lt a < b b nonngativ intgrs, and θ an irrational numbr. Thn b (nθ) min(b a, θ ). na Proof. This is straightforward computation. Th fact that b na (nθ) b a + b a is immdiat from th triangl inuality. ow, b (nθ) na (aθ) ((b + )θ) (θ) (θ) ( θ θ ) ( ) sin(πθ). It is asy to s (from thir graphs, for xampl) that sin(πx) x for all x, and th rsult follows. Lmma. Lt α b an irrational numbr, and suppos that α a for, w hav (n α) + ( + ) log. n with (a, ) and. Thn From this scond lmma w can show that if α is irrational, thn (n α) n is uidistributd (mod ): givn such α and, w hav (n α) log + n + log and sinc by Dirichlt s thorm w can tak to b arbitrarily larg, w conclud that th sunc (n α) n satisfis condition () of Wyl s critrion. As for th lmma itslf: Proof. Lt S dnot th sum in ustion. Thn S n n (α(n n )). Furstnbrg latr provd th rsult using rgodic-thortic tchnius. 4
W r-indx this sum by stting h n n so that h and, for ach such h, w hav max(, h) n n h min(, h). Th sum can thn b writtn as S min(, h) h n max( h,) (α(hn + h )) h (αh ) min(, h) n max( h,) (α(hn )). (Th its of n in this scond sum ar n whn h and h n h whn h >.) Obsrv that w hav rducd th uadratic polynomial n n to a polynomial hn + h which is linar in n, and this sum is asir to work with. This is an xampl of Wyl diffrncing. ow, using th triangl inuality and Lmma, w dduc that S h min(, hα ). Divid [, ] into intrvals of lngth at most, ach of th form h < +. W claim that th sum of min(, hα ) ovr ach such intrval is + log : W first assum that. Writ S h</ min(, hα ), and writ α a + θ with θ. Sinc h <, th rsidus of h (mod ) ar distinct, and hnc so ar th rsidus of ha (mod ) sinc (a, ). Thus ha is congrunt to ach of,,- (mod ) at most onc, and th total contribution to S in ths cass is thrfor at most. For othr valus of h, obsrv that hα ha + hθ ha h > ha >. W thus hav S + In th right-hand sid of th abov inuality, Thn / S + j h</ ah,, (mod) j ha min, ha. taks on ach of th valus,..., / / + j + log as dsird. Th cas whr h < + for othr valus of is similar. ow, thr ar clarly + of ths intrvals. It follows that j at most twic. Th rsult follows sinc S ( + log )( + ) + + ( + ) log + ( + ) log. + ( + ) log + ( + ) log. W hav shown that, for irrational α, ( n α ) is uidistributd (mod ). It follows that this sunc is n dns in [, ], and so for any ε > w can find n with n α < ε. In fact, w can us ths rsults to giv a lowr bound for an n that satisfis this inuality. Lmma. Givn and a with (a, ), thr xists m with am (log ) Proof. If > thn w can simply tak m, sinc thn am.. Thn assum. W want to 5
minimiz am, and hnc w want to find solutions in m to am b (mod ) with b small. Givn b and m, obsrv that th xprssion ( (am ) b)r r(mod) is ual to if am b(mod ), and othrwis. Thn for a givn uppr bound L on b, ϕ(l) : b L r(mod ) m ( (am ) b)r counts th numbr of solutions to am b(mod ) with b L and m. Our objctiv is thus to find a lowr bound for L subjct to th constraint that ϕ(l) >. Th contribution to ϕ(l) from r is clarly (L+). For r, th contribution to ϕ(l) is b L m ( (am ) b)r ( ) br ( am ) r b L m ( ) ( br + ) ( + ) log by Lmma. From Lmma, w hav ( ) br min L,, and sinc w hav b L b L + ( + ) log + log log. Summing ovr all r, w thus hav r ϕ(l) (L + ) log min L, r r log ( log ) (log ) i.. ϕ(l) (L+) + O( (log ) ). Hnc if L (log ) ruird. w ar guarantd solutions, as Corollary. Lt α b a ral numbr. For vry thr xists m with m α log. Proof. Lt Q b a paramtr. By Dirichlt s thorm, w can find a with Q, (a, ), and α a Q. If, thn tak m so that m α a Q Q and hnc m α Q. ow suppos >. By Lmma, thr xists m with m a (log ). Sinc m α m a m Q, w hav m α m a m Q and so m α a m + m Q (log ) + Q Q(log Q) + Q. 6
Thus, in ithr cas, w can achiv th bound m α Q(log Q) + Q. Tak Q 4 log, so that as ruird. m α log ( ) log 4 log log ( 4 log log log log + log ) + log This corollary shall b usd latr as part of a dnsity incrmnt argumnt to prov Gowrs s Thorm for th cas k 4: Thorm (Gowrs s Thorm). Thr xists a positiv constant c k such that any subst A in [, ] with A contains a non-trivial k trm artihmtic progrssion. (log log ) c k Rfrncs [] K. Chandraskharan, Introduction to Analytic umbr Thory. Springr-Vrlag, 968. [] K. Soundararajan, Additiv Combinatorics: Wintr 7. Stanford Univrsity, 7. [] S. Hartman, Rmarks on uidistribution in non-compact groups. Compositio athmatica (tom 6, p. 66-7), 964. 7