ROTH S THEOREM ON ARITHMETIC PROGRESSIONS

Transcription

1 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS ADAM LOTT ABSTRACT. The goal of this paper is to present a self-contained eposition of Roth s celebrated theore on arithetic progressions. We also present two different stronger versions of Roth s theore for two different notions of optial sets. CONTENTS. Introduction.. History of the proble.2. Notation 3 2. Preliinaries 3 3. Proof of Roth s Theore Sall Fourier coefficients Large Fourier coefficients Copleting the proof via density increent arguent 0 4. Roth s Theore for Sale sets Stateent and proof of density threshold Eaple of Sale set 5. Roth s Theore for U 2 -optial sets Preliinaries Stateent and proof of density threshold 6 References 7. INTRODUCTION.. History of the proble. A central question in additive nuber theory is to ask what conditions need to be iposed on a subset of the integers to guarantee that it contains an arithetic progression. A natural first step is to guess that if a set is big enough, it will contain an arithetic progression. In order to talk about the size of subsets of the integers in a eaningful way, we define the notion of upper density for a subset of the integers. Definition. (Upper density). The upper density of a set A Z is defined as li sup N! #(A \ [,N]). (.) N Date: April 29, 207.

2 2 ADAM LOTT Reark.2. Lower density can be defined in the sae way, with li inf replacing li sup. If the upper and lower densities of a set are equal, the set is said to have an asyptotic density. One of the first results along these lines was proven by van der Waerden in 927 [vdw], who showed that for any positive integers r and k, there eists an N(r, k) such that for any partition of {, 2,...,N} into r distinct color classes, there eists a onochroatic arithetic progression of length k. In 936, Erdős and Turán [ET] ade the stronger conjecture that any set of integers with positive upper density contains arbitrarily long arithetic progressions. This conjecture was resolved by Roth in 953 [Ro] for progressions of length three, and that result is the focus of this paper. In 969, Szeerédi [Sz] etended the result to progressions of length four. Roth used analytic ethods, while Szeerédi used an intricate cobinatorial arguent. In 972, Roth [Ro2] etended his analytic ethod to work for four-ter progressions, and in 975 Szeerédi [Sz2] etended his cobinatorial ethod to resolve the conjecture for arbitrarily long progressions. The affirative answer to Erdős and Turán s original conjecture is now known as Szeerédi s Theore. Many different proofs of Szeerédi s Theore have been produced, including an ergodic theoretical proof due to Furstenberg, Katznelson, and Ornstein [FKO]. In the late 990s, Tiothy Gowers developed new analytic achinery to work for four-ter progressions [Go] and arbitrarily long progressions [Go2], work for which he was awarded a Fields Medal. The theore has also been generalized to cases where the abient set is soe other subset of the integers. For eaple, Ben Green [Gr] showed that any set with positive upper relative density within the prie nubers contains a three-ter progression, and Green and Tao later etended this to arbitrarily long progressions [GT]. More recently, attention has been given to trying to reduce the density threshold. Roth s original work actually showed that #(A \ [,N]) c (.2) N log log N for soe constant c is sufficient to guarantee the eistence of a three-ter progression in A. This bound has been slowly lowered over the years by Heath-Brown [He], Bourgain [Bo], and Sanders [Sa]. The current record is due to Bloo [Bl], who has shown that #(A \ [,N]) c(log log N) 4 (.3) N log N suffices. The largest constructed eaple of a set containing no three-ter progressions is due to Behrend [Be] and satisfies #(A \ [,N]) ep( c p log N), (.4) N so the gap between the best known upper and lower bounds is still quite large. It is another conjecture of Erdős [Er] that any set A such that n = (.5) n2a contains arbitrarily long arithetic progressions. This conjecture is still wide open and Erdős hiself offered a $3000 reward for a solution.

3 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS 3 For the reainder of the paper, we focus our attention on Roth s original theore. Theore.3 (Roth). Let A be a subset of Z with positive upper density. contains a three ter arithetic progression. Then A The theore is often phrased in the following equivalent for, which is easier to work with. Theore.4 (Roth, finitary for). For every > 0, there eists an N 0 ( ) such that for every N N 0 and every A {, 2,...,N} with #A N, A contains a three ter arithetic progression..2. Notation. We denote by Z N the additive cyclic group Z/N Z. Throughout the paper, we identify sets with their characteristic functions in the sense that for any set S, we define the function S() :=if 2 S and S() :=0if 62 S. We also write #S for the cardinality of a finite set S. We write [N] to denote the set {, 2,...,N} Z. Also, fro now on, if the range of suation is not specified, it is understood to be all of Z N (N will always be clear fro contet). We denote by the additive character (t) :=ep(2 it/n). 2. PRELIMINARIES We will ake heavy use of the following well-known identity. Proposition 2.. We have Proof. Let 2Z N (u) = ( 0 u 6= 0 N u =0. (2.) S = (0u)+ (u)+...+ ((N )u) = + (u)+...+ ((N )u). (2.2) Then (u)s = (u)+...+ ((N )u)+ = S, (2.3) which iplies that either S =0or (u) =, which happens if and only if u =0. When u =0the su is obviously equal to N, so the proof is coplete. Our ain tool throughout will be the Fourier transfor, which is defined as follows. Definition 2.2 (Fourier Transfor). Given a function f : Z N! C, its Fourier transfor b f is defined by bf() := N 2Z N ( )f(). (2.4) The Fourier transfor has any useful properties. The following theore suarizes a few of the which will be ost useful for proving Roth s Theore. Theore 2.3. Let f : Z N! C. The Fourier transfor f b possesses the following properties. (a) Inversion forula. For any 2 Z N, f() = 2Z N () f(). b (2.5)

4 4 ADAM LOTT (b) Pointwise estiate. For any 2 Z N, we have b f() apple N (c) Plancherel s identity. 2Z N b f() 2 = N 2Z N f(). (2.6) 2Z N f() 2. (2.7) (d) Convolution identity. For f, g : Z N! C, define the convolution (f g)() := Then for any, Proof. (a) We have, by Proposition 2., () f() b = N,y y2z N f(y)g( y). (2.8) \ (f g)() = N b f()bg(). (2.9) = N y () ( y)f(y) (2.0) f(y) (( y)) (2.) (b) By the triangle inequality, b f() apple N = f(). (2.2) ( ) f() = N f(). (2.3) (c) We have f() b 2 = N 2 ( )f() ( y)f(y) (2.4),,y = N 2 f()f(y) ((y )) (2.5),y = N f() 2. (2.6) (d) We have \ (f g)() = N = N,y = N,y ( )(f g)() (2.7) ( + y y)f(y)g( y) (2.8) ( y)f(y) ( ( y))g( y) (2.9) = N y ( y)f(y) z ( z)g(z) (2.20)

5 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS 5 = N b f()bg(). (2.2) 3. PROOF OF ROTH S THEOREM The general strategy of the proof is as follows. If the Fourier coefficients b A() are sall for all 6= 0, then we can find any arithetic progressions in A by a siple counting arguent. On the other hand, if there is an 6= 0such that b A() is large, then we can find a long arithetic progression P such that the density of A \ P in P is greater than the density of A in [N]. Then, since arithetic progressions are preserved under affine transforations, we can identify P with [#P ] and repeat the process with A\P as a subset of P. Since the density increases each tie, eventually the density of A in an arithetic progression becoes greater than, which is ipossible. So we ust eventually reach a point where all of the Fourier coefficients are sall, iplying the eistence of arithetic progressions inside A. The philosophy behind this approach is that the Fourier transfor of the indicator function of a set encodes arithetic inforation about that set. Specifically, if all of the Fourier coefficients are sall, then the set is approiately rando and thus likely to contain an arithetic progression. And if a Fourier coefficient is large, it indicates that the set has ore structure along certain arithetic progressions. The proof presented here follows the proof given in [IMS] with the eception of the density increent construction, which is adapted fro [Ly]. Let A [N] and let = (#A)/N. For technical reasons later, we will want 2 to be an invertible eleent in Z N, so we assue that N is odd. This does not cause any probles because if N is even, then the density of A in [N +]is only negligibly different fro the density of A in [N]. We will consider A as a subset of Z N. The advantage of this is that we now have all of the tools of Fourier analysis at our disposal. The downside is that arithetic progressions in Z N are not necessarily progressions in Z (they ight wrap around ), but this will be taken care of. We will soeties refer to a three-ter arithetic progression as a 3AP, and if in Z N is not specified, then we ean a 3AP in Z. Note that {, y, z} is a 3AP in Z N if + z 2y (od n). To guarantee that we only count 3APs in Z, we define B := A \ [N/3, 2N/3] and note that if, y 2 B, then {, y, z} is a 3AP in Z. Denoting the nuber of 3APs contained in A by Q, we can count Q #{(, y, z) 2 B B A : + z 2y (od n)} (3.) = B()B(y)A(z) (3.2),y,z, +z 2y = N B()B(y)A(z) ( ( + z 2y)) (3.3),y,z = N B() ( ) B(y) (2y) A(z) ( z) (3.4) y z = N 2 bb() b B( 2) b A() (3.5)

6 6 ADAM LOTT = N (#B) 2 (#A)+N 2 6=0 bb() b B( 2) b A() (3.6) = N (#B) 2 (#A)+E. (3.7) At this point, we ay assue that #B #A/5. If this is not the case, then either A \ [0,N/3] or A \ [2N/3,N ] ust have size at least 2(#A)/5 and hence has relative density at least 6 /5 > in its abient progression. Thus the density increent arguent kicks in (i.e., we ay replace A by A \ [0,N/3] and [N] by [N/3] and repeat the sae arguent). 3.. Sall Fourier coefficients. When all of the nonzero Fourier coefficients b A() are sall, we can use (3.7) to establish the eistence of a three-ter progression directly. Theore 3.. If b A() < 2 /0 for all 6= 0, then A contains a three ter arithetic progression. Proof. We prove this by showing that the error ter E in (3.7) is sall. By the Cauchy- Schwartz inequality and Plancherel s identity, we have E apple N 2 a A() b bb() B( b 2) (3.8) 6=0 apple N 2 a A() b bb() b B( 2) (3.9) apple N 2 a 6=0 b A() = N a 6=0 b A()! /2 B() b 2! /2 B() 2! /2 B( b 2) 2! /2 B( 2) 2 (3.0) (3.) apple N 2 0 (#B) apple 2 N (#B) 2 (#A) (3.2) because of our assuption that #B Q (#A)/5. This together with (3.7) iplies that 2 N (#B) 2 (#A) 3 50 N 2. (3.3) Since the count Q does not eclude the trivial progression = y = z, we have to subtract those off, but there are only #A = N of those, so the nuber of nontrivial 3APs contained in A is at least which is positive if N is sufficiently large N 2 N, (3.4)

7 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS Large Fourier coefficients. If A has a large Fourier coefficient, then the estiate given by (3.7) is not useful, so we ust use the additional arithetic inforation that the large Fourier coefficient encodes. In this section, it will be convenient to work with the so-called balanced function f() :=A(). We quickly describe the iportant properties of f. Proposition 3.2. The balanced function f possesses the following properties. (a) P f() =0. (b) f() b = A() b for all 6= 0and f(0) b = 0. Proof. We have and bf() =N f() = A() = #A N = 0 ( )f() =N ( )A() N ( ) = b A() if 6= 0. Also, bf(0) = N f() = 0. In this section we will prove the following theore, which is the stateent that if A has a large Fourier coefficient, then there is a long arithetic progression P on which A has increased density. Theore 3.3. Suppose that b A(r) > > 0 for soe r 6= 0. Then there eists an arithetic progression P satisfying #P ( /64) p N and #(A\P ) ( + /8)(#P ). The proof consists of three basic steps. Step : Construct a long Z N -arithetic progression P such that b P (r) is also large. Step 2: Show that this iplies that A has a large intersection with soe translate of P, call it P 0. Step 3: Lift the Z N -progression P 0 to a Z-progression P 00 without losing too uch length or too uch intersection with A. Proof of Theore 3.3. Step. Suppose that b A(r) > for soe r 6= 0. Consider the set of points {(0, 0), (,r), (2, 2r),...,(N, (N )r)} [N ] 2. By subdividing [N ] 2 into a b p N c b p N c grid, there are fewer than N total boes so the pigeonhole principle iplies that two points are in the sae bo, i.e., there eist p, q such that both p q apple p N and r(p q) apple p N when considered odulo N (see Figure ).

8 8 ADAM LOTT FIGURE. An eaple of the above construction with N =, r =4, b p N c =3. Notice that there are two points in the botto left and top right boes. Let d = p q. Let P be the progression {..., d, 0,d,...} of length b p N/8c where the endpoints of P are chosen so that it is as syetric around 0 as possible. We want b P (r) to be large, so consider N b P (r) #P = (P () ( r) P ()) (3.5) apple ( r) (3.6) 2P apple (rd`) (3.7) ` apple(#p )/2 apple #P a (rd`). (3.8) ` apple(#p )/2 By construction, we have that rd` apple p N (#P )/2 apple N/6. Thus, (rd`) is an Nth root of unity which is no ore than /6th of the way around the unit circle. Hence for any `, (rd`) apple 2 /6 < /2. Thus we conclude that so that b P (r) > (/2)N (#P ). N bp (r) #P < 2, (3.9) Step 2. We now want to show that this iplies that A has a large intersection with soe translate of P. Define G() :=(f P )() = P y f(y)p ( y) where f is the

9 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS 9 balanced function of A. Note that G has ean value zero because G() = f(y)p ( y) = f(y) P ( y) = (#P )f(y) = 0.,y y y (3.20) By the pointwise estiate and the convolution identity of Theore 2.3, we have N G() b G(r) = N b f(r) b P (r) (#P ). (3.2) 2 Since G has ean value zero, we can write the above as G() + G() N (#P ), (3.22) 2 so that G() + G() (#P ) for soe, which iplies that G() (#P ). 2 4 Epanding out definitions, we get so that 4 (#P ) apple f(y)p ( y) (3.23) y = A(y)P ( y) P ( y) (3.24) y y = #(A \ ( P )) (#( P )) (3.25) #(A \ ( P )) ( + )(#( P )). (3.26) 4 Step 3. Let P 0 = P. The only reaining obstacle is that although P 0 is an arithetic progression in Z N, it ay not be an arithetic progression in Z. We need to find an arithetic progression in Z contained in P 0 that is still relatively long and still has a relatively large intersection with A. We accoplish this in the following way. Note that since #P 0 apple p N and the coon difference of P 0 is also at ost p N, the last ter of P 0 does not wrap around the first ter of P 0 odulo N. Hence P 0 can be written as P 0 = P [ P 2 where both P and P 2 are arithetic progressions in Z. Without loss of generality, suppose that #P apple #P 2. If #P apple (/8) (#P 0 ), then #(A \ P 2 ) #(A \ P 0 ) #P (3.27) ( + 4 )(#P 0 ) 8 (#P 0 ) (3.28) = ( + 8 )(#P 0 ). (3.29) If, on the other hand, #P (/8) (#P 0 ), then both P and P 2 have length at least (/8) (#P 0 ) and since P 0 = P [ P 2, A ust have density of at least +(/4) on one of the. Thus we have established the eistence of an arithetic progression P 00 in Z such that #P 00 (/8) (#P 0 ) and #(A \ P 00 ) ( +(/8) )(#P 00 ). Since #P 0 = b p N/8c, this copletes the proof.

10 0 ADAM LOTT 3.3. Copleting the proof via density increent arguent. Suppose A is a subset of [N] containing no 3APs. By Theore 3., this iplies that for soe r 6= 0, A(r) b 2 /0. Then by Theore 3.3 with = 2 /0, there eists an arithetic progression P such that #P ( 2 /640) p N and #(A \ P ) ( + 2 /80)(#P ). Let A = A \ P. Since arithetic progressions are preserved under affine transforations, we can identify P with [#P ] and A with a subset of [#P ] of density + 2 /80. Since we assued A contains no 3APs, obviously neither does A, so we ay repeat the arguent and obtain another progression P 2 and A 2 := (A \ P 2 ) where the density of A 2 in P 2 is 2 + /80. 2 Repeating this process, we get a sequence of progressions P k and subsets A k with relative densities k satisfying 2 #P k p 2 k k #Pk, k k (3.30) Notice that k + k( 2 /80). Thus after k =80/ iterations, we have k 2. Now for k>80/, we have k 2 +k((2 ) 2 /80), so that after k =80/(2 ) ore iterations, the density has increased fro 2 to 4. In general, the density will have increased to 2 a after no ore than (80/ )(+/2+/4+...+/2 a ) apple 60/ iterations. Picking a to be sufficiently large, we see that the density has increased past after a finite nuber of iterations, which is a contradiction. This copletes the proof of Roth s Theore. Reark 3.4. This ethod also yields an upper bound for the constant N 0 ( ) entioned in Theore.4. We have shown that at ost k =60/ iterations are needed to arrive at a contradiction, so N 0 just needs to be large enough so that #P k > 0 after k =60/ iterations (see equation (3.30)). In [Ly], it is shown that suffices for soe constant C>0. N 0 ( ) = ep(ep(c )) 4. ROTH S THEOREM FOR SALEM SETS 4.. Stateent and proof of density threshold. In the proof of Roth s Theore, it becoes clear that when all of the nonzero Fourier coefficients A() b are sall in absolute value, it is very easy to establish the eistence of three ter arithetic progressions contained in A. This suggests the following question. If the Fourier coefficients of A are optially sall, can we get away with a saller density threshold to guarantee the eistence of three ter arithetic progressions contained in A? To answer this question, we first need to understand how sall is optially sall. Plancherel s identity tells us A() b 2 = N A() 2 = N (#A), (4.) which tells us that, even in theory, the sallest order of agnitude for b A() that we can hope for is N p #A. This observation leads to the following definition. Definition 4. (Sale set). Let {A N } N= be a faily of sets with A N Z N. The faily is said to be Sale if there eists a universal constant C such that for all N and all nonzero 2 Z N, A c N () apple CN p #A N. (4.2)

11 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS For Sale sets, it is possible to guarantee the eistence of three ter arithetic progressions when the density is uch saller. Since the Sale condition only akes sense within the fraework of Z N, for this section and the net, we will only consider arithetic progressions in Z N. The arguent given here is presented in [IMS]. Theore 4.2. Let {A N } N= be a Sale faily. If there eists a universal constant c such that #A N cn 2/3, then for all sufficiently large N, A N contains a three ter arithetic progression. Proof. Let Q N (t) :={, y, z 2 A N : y = z y = t} be the nuber of three ter arithetic progressions in A N with coon difference t. The nuber of three ter arithetic progressions contained in A N is then equal to Q N (t) = A N ( t)a N ()A N ( + t) (4.3) t,t =,t A N () (( t)) c A N () ` (( + t)`) c A N (`) (4.4) =,,` A N () c A N () c A N (`) (( + `)) t (t(` )) (4.5) = N, A N () (2) c A N () c A N () (4.6) = N 2 ca N () c A N () c A N ( 2) (4.7) = N (#A N ) 3 + N 2 6=0 c AN () c A N () c A N ( 2) (4.8) = N (#A N ) 3 + E (4.9) To bound the error ter E, we can use the Sale condition to get E apple N 2 6=0 c A N () 2 c A N ( 2) apple C 3 (#A N ) 3/2. (4.0) If c is sufficiently large with respect to C and #A N cn 2/3, then (4.0) iplies that E apple (/2)N (#A N ) 3. Hence the nuber of nontrivial three ter arithetic progressions in Z N is at least 2 N (#A N ) 3 (#A N ) (#A N )((/2)N (#A N ) 2 ) (4.) which is positive if N is sufficiently large. cn 2/3 ((c 2 /2)N /3 ) (4.2) 4.2. Eaple of Sale set. At first glance, it is not obvious that Sale sets actually eist. In this section, we provide an eaple of a Sale faily of subsets of Z p as p ranges over the pries. First we need a lea. Lea 4.3. For any prie p and any a 6= 0, let G(a) = P 2Z p (a 2 ). Then we have G(a) = p p. (4.3)

12 2 ADAM LOTT Proof. We have G(a) 2 =,y2z p (a 2 ) (ay 2 ) =,y2z p (a( 2 y 2 )). (4.4) The change of variables t = y, u = + y is bijective and tu = 2 y 2, so we have G(a) 2 = t,u2z p (atu) (4.5) = r2z p t,u tu=r (ar) (4.6) = r2z p (ar)(r) (4.7) where (r) :=#{(t, u) 2 Z 2 p : tu = r}. We know tu =0if and only if t =0or u =0, so (0) = 2p. If r 6= 0, then t can be any nonzero eleent and u is deterined, so (r) =p. Hence G(a) 2 = (0) + (ar)(r) (4.8) r6=0 = 2p +(p ) r6=0 (ar) (4.9) = 2p (p ) + (p ) r (ar) (4.20) = p. (4.2) Theore 4.4. Define E p := {t 2 : t 2 Z p }. Then {E p } is a Sale faily. Proof. For any 6= 0, we have ce p () = p ( )E p () = p 2E p ( ). (4.22) Since eactly half of the nonzero residue classes in Z p are squares and every nonzero square has eactly two square roots, this becoes ce p () = p (0) + 2 p t6=0 = p (0) + 2 p t = 2 p (0) + t ( t 2 ) (4.23) ( t 2 ) ( t 2 )! 2 p (0) (4.24). (4.25) By Lea 4.3, we have ce p () apple 2 p ( + p p) apple p /2. (4.26)

13 Since #E p =(p +)/2, we have ROTH S THEOREM ON ARITHMETIC PROGRESSIONS 3 p p #E p 2 p /2 E 2 c p () (4.27) for every 6= 0, so {E p } is a Sale faily. 5. ROTH S THEOREM FOR U 2 -OPTIMAL SETS 5.. Preliinaries. In the previous section, we defined a notion of what it eans for a set to be optially rando and showed that sets satisfying that condition require a uch saller density threshold in order to guarantee the eistence of a three-ter progression. In this section, we will introduce a weaker notion of randoness and show that sets with this condition also get away with a saller density threshold to guarantee the eistence of a three-ter progression. The relevant concepts were first introduced by Gowers in [Go2] in his proof of Szeerédi s Theore. Definition 5. (L p nor). For any function f : Z N! C and any p>0, the L p nor of f is defined as f p :=! /p f() p. (5.) Definition 5.2 (Gowers unifority nor). For any function f : Z N! C, the Gowers U 2 nor is defined by the forula f U 2 := N! /4 3 f()f( + h )f( + h 2 )f( + h + h 2 ). (5.2),h,h 2 Reark 5.3. The U 2 nor is the nor that is best suited for handling three-ter progressions. In fact, Gowers introduced U k nors for all k 2 which are ore suitable for handling longer progressions. However, the difficulty of the arguent increases draatically for k 3. Reark 5.4. The U 2 nor satisfies all the necessary properties of a nor and even satisfies a generalized Cauchy-Schwartz inequality, but we never need to use these facts, so we will not prove the. The U 2 nor has an interpretation in ters of the Fourier transfor, as the following proposition shows. This is otivation for why it is useful for studying three-ter progressions. The reason that the cases of longer progressions are so uch harder is that for k>2 the U k nor no longer has a nice Fourier analytic interpretation. Proposition 5.5. For any function f : Z N! C, we have Proof. We have f 4 U 2 = N 3 f U 2 = b f 4. (5.3),h,h 2 f()f( + h )f( + h 2 )f( + h + h 2 ) (5.4)

14 4 ADAM LOTT = N 3 a,b,c,d a+d=b+c = N 4 f(a)f(d)f(b)f(c) (5.5) ((b + c (a + d))) (5.6) f(a)f(d)f(b)f(c) a,b,c,d = N 4 f(a) ( a)f(d) ( d)f(b) ( b)f(c) ( c) (5.7) = a,b,c,d b f() 4 = b f 4 4. (5.8) In order to prove anything about optial sets, it is first necessary to deterine how sall the U 2 nor of a characteristic function can be. Since the U 2 nor is related to the Fourier coefficients, we can observe that for any set A, so that A 4 U 2 = b A() 4 = (#A) 4 N 4 + 6=0 b A() 4 (5.9) 2 A 4 U 2 (#A)4 N sup 4 A() b A() b 2 (5.0) 6=0 6=0 2 sup A() b A() b 2 (5.) 6=0 2 = sup A() b N A() 2 (5.2) 6=0 2 = sup A() b N (#A). (5.3) 6=0 Recall fro the Sale condition (which was deduced directly fro Plancherel s identity) that sup 6=0 b A() can never be saller than (#A) /2 N, so we are left with A 4 U 2 (#A)4 N 4 (#A) 2 N 3. (5.4) At this point, it will be useful to recall the definition of the balanced function f() :=A() (#A)N introduced in Section 3.2. The following lea relates the nors of the characteristic function A and its balanced function f. Lea 5.6. Let A be a subset of Z N, let =#A/N, and let f be the balanced function of A. Then f 4 U = 2 A 4 4 U. (5.5) 2 Proof. We have A 4 U 2 = N 3,h,h 2 A()A( + h )A( + h 2 )A( + h 3 ) (5.6)

15 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS 5 = N 3,h,h 2 ( + f())( + f( + h ))( + f( + h 2 ))( + f( + h + h 2 )) = N 3 ",h,h 2 (5.7) # 4 + f()f( + h )f( + h 2 )f( + h + h 2 )+(other ters) where each of the ters in (other ters) is a product of the for (5.8) g ()g 2 ( + h )g 3 ( + h 2 )g 4 ( + h + h 2 ) (5.9) where at ost three of the g i are equal to f and the others are equal to. Thus, by changing variables, since f has ean value zero (recall Proposition 3.2), each of the ters in (other ters) vanishes. So we have A 4 U = 4 + N 3 f()f( + h 2 )f( + h 2 )f( + h + h 2 ) (5.20),h,h 2 = 4 + f 4 U 2. (5.2) Equation (5.4) and Lea 5.6 otivate the following definition. Definition 5.7 (U 2 -optial set). Let {A N } N= be a faily of subsets of Z N and let f N be the balanced function of A N. We say the faily is U 2 -optial if there eists a universal constant C such that f N 4 U 2 = A N 4 U 2 (#A)4 N 4 apple C(#A) 2 N 3. (5.22) We also prove here one ore lea, adapted fro [Gr2], which will be necessary to prove the density threshold for U 2 -optial sets. Lea 5.8. Let f : Z N! [, ] be any function. Then f(y)f(y + d)f(y +2d) apple N 2 f U 2. (5.23) y,d Proof. Let Q denote the quantity P y,d f(y)f(y + d)f(y +2d). We have, by the change of variables = 2(y + d) and repeated application of the Cauchy-Schwartz inequality, Q =,y f(y)f(/2)f( y) (5.24) apple! 0 /2 f(/2) f(y)f( y) y 2 A/2 (5.25) apple N /2 /2 f(y)f( y)f(z)f( z)! (5.26),y,z

16 6 ADAM LOTT 2 apple N /2 6 4 f(y) 2 apple N /2 4N /2 y 2 y apple N 3/4 4 y apple N 3/4 N /4 = N N a+b=c+d z f(y) f(y) 2! /2,y,z,t = N 2 N 3 a+b=c+d = N 2 N 3! 0 /2 f(z) z 3 2 /2 /2 f( y)f( z) A 7 5! 3 /2 f( y)f( z)f(t y)f(t z) 5,z,t y /2 (5.27) (5.28)! 3 /2 f( y)f( z)f(t y)f(t z) 5,z,t (5.29) f( y)f( z)f(t y)f(t z)! /4 (5.30) f(a)f(b)f(c)f(d)! /4 (5.3) f(a)f(b)f(c)f(d)! /4 (5.32),h,h 2 f()f( + h )f( + h 2 )f( + h + h 2 )! /4 (5.33) = N 2 f U 2. (5.34) 5.2. Stateent and proof of density threshold. We now have the proper notions in place to state the ain theore of this section. Theore 5.9. Let {A N } N= be a U 2 -optial faily of sets. There eists a constant c such that if #A N cn 9/0, then for all sufficiently large N, A N contains a three-ter arithetic progression. Proof. Let Q N := P,d A N()A N ( + d)a N ( +2d) denote the nuber of three-ter progressions contained in A N. Also let =#A/N and let f N be the balanced function of A N. We have Q N =,d ( + f N ())( + f N ( + d))( + f N ( +2d)) (5.35) /2 = N 2 3 +,d f N ()f N ( + d)f N ( +2d)+(other ters) (5.36) where, as in Lea 5.6, each of the ters in (other ters) is a product g ()g 2 ( + d)g 3 ( +2d) (5.37)

17 ROTH S THEOREM ON ARITHMETIC PROGRESSIONS 7 where at ost two of the g i are equal to f N and the others are equal to. So again, every ter in (other ter) vanishes when sued over and d. So we have by Lea 5.6, Lea 5.8, and the U 2 -optial hypothesis that Q N N 2 3 =,d f N ()f N ( + d)f N ( +2d) (5.38) apple N 2 f N U 2 (5.39) apple CN 2 ((#A N ) 2 N 3 ) /4 (5.40) This shows that Q N is guaranteed to be at least (/2)N 2 3 =(/2)(#A N ) 3 N provided that CN 2 (#A N ) /2 N 3/4 < (/2)N 2 3. This happens if #A N > (2C) 2/5 N 9/0, (5.4) so for any A N satisfying (5.4), the nuber of nontrivial three-ter progressions it contains is at least 2 (#A N) 3 N (#A N ) = (#A N ) 2 (#A N) 2 N, (5.42) which is positive for sufficiently large N. REFERENCES [Be] F. Behrend. On the sets of integers which contain no three in arithetic progression. Proc. Nat. Acad. Sci. 23 (946), [Bl] T. F. Bloo. A quantitative iproveent for Roth s theore on arithetic progressions. J. London Math. Soc. 93 (206), [Bo] J. Bourgain. Roth s theore on progressions revisited. J. Anal. Math. 04 (2008), [Er] P. Erdős. Probles in nuber theory and cobinatorics. Proc. 6th Manitoba Conference on Nuerical Math. (976), [ET] P. Erdős and P. Turán. On soe sequences of integers. J. London Math. Soc. (936), [FKO] H. Furstenberg, Y. Katznelson, and D. Ornstein. The ergodic theoretical proof of Szeerédi s [Go] Theore. Bull. Aer. Math. Soc. 7, no. 3 (982), W. T. Gowers. A new proof of Szeerédi s Theore for arithetic progressions of length four. Geo. Funct. Anal. 8 (998), [Go2] W. T. Gowers. A new proof of Szeerédi s Theore. Geo. Funct. Anal. (200), [Gr] B. Green. Roth s Theore in the pries. Ann. Math. 6 (2005), [Gr2] B. Green. Montreal Lecture Notes on Quadratic Fourier Analysis. Lecture notes. [GT] B. Green and T. Tao. The pries contain arbitrarily long arithetic progressions. Ann. Math. 67 (2008), [He] D. R. Heath-Brown. Integer sets containing no arithetic progressions. J. London Math. Soc. 35 (987), [IMS] A. Iosevich, A. Mayeli, and S. Senger. Fourier bases: an eleentary viewpoint on a variety of applications. In preparation. [Ly] N. Lyall. Roth s Theore, the Fourier analytic approach. Lecture notes. [Ro] K. Roth. On certain sets of integers. J. London Math Soc. 28 (953), [Ro2] K. Roth. Irregularities of sequences relative to arithetic progressions. IV. Period. Math. Hungar. 2 (972), [Sa] T. Sanders. On Roth s theore on progressions. Ann. Math. 74 (20),

18 8 ADAM LOTT [Sz] E. Szeerédi. On sets of integers containing no 4 eleents in arithetic progression. Acta Math. Acad. Sci. Hungar. 20 (969), [Sz2] E. Szeerédi. On sets of integers containing no k eleents in arithetic progression. Acta Arith. 27 (975), [vdw] B. L. van der Waerden. Beweis einer Baudetschen Verutung. Nieuw Arch. Wisk. 5 (927),