AN EASY INTRODUCTION TO THE CIRCLE METHOD

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1 AN EASY INTRODUCTION TO THE CIRCLE METHOD EVAN WARNER Thi talk will try to ketch out oe of the ajor idea involved in the Hardy- Littlewood circle ethod in the context of Waring proble.. Setup Firt, let etablih a general etup. We could trive for ore generality, but thi fraework will allow u to dicu any of the proble that fall under the purview of the circle ethod. Let A be a ubet of the natural nuber N (here conidered o a to exclude zero), and et r(n;, A) : #{way to write n a a u of eleent of A}. Many iportant proble can be phraed in ter of thee function. For exaple, let A {2, 3, 5, 7,...} be the et of prie nuber. Then ternary Goldbach conjecture r(n; 3, A) > for all n odd and greater than 5 and binary Goldbach conjecture r(n; 2, A) > for all n even and greater than 2. A another exaple, let A {, 2 k, 3 k, 4 k,...} be the et of kth power, where k i an integer 2. Then, being deliberately vague, Waring proble knowledge of r(n;, A). There are other proble that can be attacked with the circle ethod but that don t quite fit into our fraework here; for exaple, deterining ayptotic of the partition function p(n) or proving theore about all gap in prie. For thi talk, we ll concentrate on Waring proble, a it i of oderate difficulty and illutrate oe baic idea nicely. There will be no effort at coplete proof. 2. Background on Waring proble Fix an integer k 2 for the reainder of the talk. We re intereted in the nuber of way of expreing a poitive integer a a u of kth power. If A i the et of kth power, et r (n) : r(n;, A). Waring original proble wa to prove that for all k, there exit an uch that r (n) > for all n >. Thi proble wa proven by Hilbert in 99, uing idea of Hurwitz that predate the circle ethod. Of coure, refineent iediately preent theelve. What i the inial uch, a quantity uually denoted by g(k)? Forula are now known for g(k), due to variou 2th century atheatician. More interetingly (and ore challengingly), one can ak what i the iniu if we allow finitely any exception in n, a quantity denoted by G(k)? Here I believe the bet known bound i G(k)

2 2 EVAN WARNER k(log k + log log k C log log k/ log k) for oe contant C, due to Wooley. It i conjectured that G(k) hould be approxiately linear in k. Finally, we can ak for ayptotic for r (n). We will exaine thi proble, following Hardy-Littlewood, for ufficiently large in ter of k. Specifically, we will require 2 k + ; the larget range on which the ayptotic we will derive are known to be valid i k 2 (log k + log log k + C) for oe contant C, due to Ford. The conjectured range i the coniderably tronger k Generating function Our firt idea i that generating function are fun and ueful, o let ake a generating function for the characteritic function of our et A. Set F (x) : x a x jk. a A Then we ee iediately that F (x) j x jk j r (n)x n, jut by regrouping ter. Thu F (x) i the generating function for r (n). Now, to pick out the coefficient, we integrate around a circle (hence the nae circle ethod ). Let γ be a counterclockwie circle centered at the origin of radiu r. Totally ignoring convergence iue for the oent, we calculate that 2πi γ F (x) dx xn+ 2πi j n r (j)x j n dx γ j r (j) 2πi γ x i n dx r (n), uing the well-known fact fro baic coplex analyi that 2πi γ x dx i equal to one if and zero otherwie. Thi calculation turn out to be valid when r <. Thu, the proble of finding r (n) turn into the evaluation or etiation F (x) of the integral 2πi x dx. In fact, thi i what Hardy and Littlewood did. n+ γ 4. Generating function à la Vinogradov We are going to do oething lightly different, following a technical refineent due to Vinogradov: intead of uing a power erie generating function and integrating over a circle, we ue a trigonoetric erie and integrate over the line egent [, ]. Set e(x) : e 2πix. Again ignoring convergence iue (thi tie they re ore eriou, a what I going to write down doen t actually ake very uch ene), define f(x) : e(ax) e(j k x). a A j

3 AN EASY INTRODUCTION TO THE CIRCLE METHOD 3 Then, a before, we find that f(x) e(j k x) j r (n)e(nx) i the correponding trigonoetric generating function for r (n). We can eaily calculate f(x) e( nx) dx r (j)e(jx)e( nx) dx j r (j) j r (n), n e(jx)e( nx) dx where we ued that e(jx)e( nx) dx i equal to one if n j and zero otherwie. Now, we introduce in one woop a iplification and a fix to our convergence iue. Let P f P (x) : e(j k x), j where P i a finite poitive integer. Then f P (x) r (n)e(nx), where n r (n) #{olution to x k x k n uch that x i P for each i}. Thi i, of coure, very uch not the function we were looking for. But wait! A long a P n /k we do have r (n) r (n), o in that range f P (x) f(x). So to find ayptotic for r (n) it uffice to calculate f P (x) e( nx) dx for P n /k, and thi integral converge nicely. 5. Splitting up into ajor and inor arc To etiate thi integral well, let think a bit about exponential u. If we have a collection of N rando nuber on the unit circle and add the up, we expect the u to be about N (thi i an intance of the quare root law for rando walk). For ufficiently generic x [, ], the collection of nuber {e(j k x)} j P hould look fairly rando, at leat if P i large enough. So for ufficiently generic x, we expect f P (x) P. Thi can t be true for all x, however. For exaple, f P () P e(i k ) P, i

4 4 EVAN WARNER which in particular how that we can t expect to get anything ueful by bounding the integral by up x f P (). A little experienting how that thi lack of cancellation occur fairly often whenever x i very cloe to a rational nuber with all denoinator not alway, but enough to be worrying. For exaple, if k 2 then we find f P (/2) i actually bounded (excellent cancellation!) but f P (/8) CP for oe nonzero contant C becaue quare are biaed odulo 8: half the tie, they are equal to odulo 8, and a quarter of the tie each they are equal to and 4 odulo 8. Let foralize thi intuition a follow: fix a all δ > (if you prefer a definite value, feel free to take δ /). Define { M a,q x R/Z : x a } q < P k+δ R/Z. Each of thee i a allih interval around the fraction a/q. We let M M a,q. q P δ a q,(a,q) Thi i a union over all M a,q uch that a/q i a fraction in R/Z with all denoinator (copared to P ). We ll call M, a a et, the ajor arc. Thi i the locu on which we expect little cancellation and therefore expect the bulk of the integral to lie. Let [, ] M be the copleent, the inor arc, for which we do expect oe cancellation. Thi etup i ubiquitou in the context of the circle ethod. The goal going forward are the following: Evaluate M f P (x) e( nx) dx, and Show that f P (x) e( nx) dx i all copared to the above. 6. Major arc On the ajor arc, f P (x) i oehow alot contant. Let ake thi precie a follow: let x M a,q and write y x a q. Lea 6.. We have f P (x) q q P e( k a/q) e(yξ k ) dξ + O(P 2δ ). Proof (ketch). Firt, we collect ter i in the ae reidue cla odulo q. If we write i qj +, q, then f P (x) q e( k a/q) P/q The inner u i alot equal to the integral q P j e(yξ k ) dξ e(y(qj + ) k ). by replacing j by a continuou variable η and etting ξ qη+. The error i crudely etiated to be O(P 2δ ) i crudely etiated by calculating the derivative.

5 AN EASY INTRODUCTION TO THE CIRCLE METHOD 5 When we plug in the above lea to the integral over the ajor arc and crudely etiate (the tep are not difficult and alo not enlightening), we get the following reult: f P (x) e( nx) dx P k S(P δ, n)j(p δ ) + O(P k δ ) M for oe δ >, where and S(P δ, n) J(P δ ) q q P δ a (a,q) γ <P δ ( q q e( a/q)) k e( na/q) ( e(γξ k ) dξ) e( γ) dγ. Needle to ay, thi in t particularly pretty. The o-called ingular erie S(P δ, n) i rather tricky; it i eay enough to how that we can replace S(P δ, n) by S(n) : li X S(X, n) with acceptable error, but the convergence of thi u i a bit delicate. Fairly eleentary arguent uffice to how that it converge a long a 2 k + ; in fact, in thi cae it i bounded below (in n) by a contant. We will not dicu thi apect further. The J(P δ ) ter i eaier to deal with. It i eay to how that we can replace it with ( J e(γξ k ) dξ) e( γ) dγ with acceptable error. A fun integration exercie yield J Γ( + /k), Γ(/k) o in total we have f P (x) e( nx) dx S(n)P k J + O(P k δ ) M Γ( + /k) n /k S(n) + O(n /k δ ). Γ(/k) Becaue we expect the integral over the inor arc to be ubued into the error ter, thi i our expected anwer. 7. Minor arc and concluion To etiate the inor arc, we firt do oething crude: f P (x) e( nx) dx f P (x) dx. Reeber, we expect cancellation in f P itelf, o we don t need the help that e( nx) ight provide! Thi i a claical exponential u proble, and we have two claical tool that we black-box (neither i terribly difficult, and neither i the bet poible):

6 6 EVAN WARNER Theore 7. (Weyl). Let f R[X] with deg f k and highet coefficient α. Alo uppoe that there exit integer a and q uch that (a, q), q >, and α a q q 2. Aue that P δ q P k δ and let ɛ >. Then P e(f(i)) k,ɛ P δ/2k +ɛ. i That i, we get a tiny bit of iproveent over the trivial bound (a factor of P δ/2k+ +ɛ better) o long a α i well-approxiated by a fraction with relatively large denoinator ( P δ ), but not too large ( P k δ ). Theore 7.2 (Hua). We have f P (x) 2k dx k,ɛ P 2k k+ɛ. Again, we get a light iproveent over the trivial bound. Both proof (of Weyl inequality and of Hua ) eploy induction (on deg f and k, repectively) and ue Cauchy inequality repeatedly. Now aue 2 k +. On the inor arc, we ue Dirichlet theore on Diophantine approxiation to note that x ha a rational approxiation a q uch that P δ q P k δ and α a q q 2. Therefore we can apply Weyl inequality to yield f P (x) 2k P ( 2k )(+ɛ δ/2 k+). Uing Hua inequality in the third line, we have f P (x) dx f P (x) 2k f P (x) 2k dx P ( 2k )(+ɛ δ/2 k+ ) f P (x) 2k dx P +ɛ δ/2k+ 2 k (+ɛ δ/2 k+) P 2k k+ɛ P k δ for oe δ > n /k δ. Thi i indeed oething that we can ubue under the error ter fro the ajor arc. Therefore, putting everything together, r (n) Γ( + /k) n /k S(n) + O(N /k δ ) Γ(/k) o long a 2 k +. Becaue (a entioned before) the ingular erie i bounded below, we get the expected reult that r (n) a n whenever 2 k +. That i, the nuber of way of writing n a a u of kth-power tend to infinity a n doe, o long a 2 k +.

7 AN EASY INTRODUCTION TO THE CIRCLE METHOD 7 Heuritically, the quantity i the denity of olution to Γ( + /k) n /k Γ(/k) x k x k n where the x i are poitive and real, while S(n) i the ter that take into account congruence between kth power that ake the integer act le regularly than the real. Thi can, of coure, be ade ore precie. 8. Reference Davenport, H., Analytic ethod for Diophantine equation and Diophantine inequalitie, Cabridge (25) Vaughan, R. C. and Wooley, T. D., Waring proble: A urvey, ath.la.uich.edu/~wooley/wp.p

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