Furstenberg’s proof of long arithmetic progressions Introduction to Roth’s Theorem

Furstenberg’s proof of long arithmetic progressions:Introduction to Roth’s TheoremSteven ButlerUniversity of California,San DiegoNovember4,2004These are the notes for thefirst of a pair of lectures that will outline a proof given by Hillel Furstenberg[3]for the existence of long arithmetic progressions in sets of integers with positive upper density,a resultfirst proved by Szemer´e di[8].1History of long arithmetic progressionsThefirst major result in the theory of long arithmetic progressions was due to van der Waerden in1927[9].Theorem(van der Waerden).Given any partition of the positive integers intofinitely many sets B1,...,B k,then at least one of the B i contains arithmetic progressions of arbitrary length.In the1930’s Erd˝o s and Tur´a n[2]conjectured that for a set to have arbitrarily long arithmetic progressions that it needed to be asymptotically dense.Specifically,any set with positive upper density should have arbitrarily long arithmetic progressions.Definition(See[4],p.72).A set B is said to have positive upper Banach density if and only if|B∩I|lim sup|I|→∞The definition for positive upper density is similar,only in that case we restrict ourselves to intervals of the form[1,n].Clearly any set with positive upper Banach density has positive upper density.The proof by Furstenberg will show that any set with positive upper Banach density will have long arithmetic progressions.[Note that in van der Waerden’s theorem at least one of the B i must have positive upper density,and so the conjecture by Erd˝o s and Tur´a n is a stronger claim.]Roth in1952[6]established that any set with positive upper density contains an arithmetic progression of length three.Szemer´e di in1969[7]established the result for arithmetic progressions of length four and in1975[8]established the result for arithmetic progressions of arbitrary length using combinatorial arguments.Theorem(Szemer´e di).Any set of integers having positive upper density contains arith-metic progressions of arbitrary length.Shortly after Szemer´e di established the result,Furstenberg in1977[3]established the result by translating it into a measure theoretic question and then used ergodic theory.2A quick introduction to ergodic theoryErgodic theory deals with the study of long term average behavior of dynamical systems. The principle being that“behavior on the average is easier to control”.The setting for ergodic theory is(X,B,µ,T)where X is the space,B is aσ-algebra of sets for the space(these represent the measurable sets of X),µis afinite measure(i.e.,µ(X)<∞, by scaling we can,and will,assume thatµ(X)=1so that we are dealing with a probability measure),and T:X→X is a measure preserving transformation.Definition.We say T:X→X is a measure preserving transformation if for every A∈B we have thatµ(A)=µ(T−1A).Further,we say that a measure preserving transformation is ergodic if A=T−1(A)implies thatµ(A)=0orµ(A)=1.Note,we do not require that T be invertible,in which case we may not haveµ(A)=µ(T A). Also,for any measure preserving transformation system we have that it can be decomposed into a collection of ergodic systems in a manner analogous to the fact that any permutation can be replaced by a product of cycles.Thefirst result in ergodic theory can be traced back to Poincar´e,which we will talk about below in relationship to two term arithmetic progressions.Birkhoffin1927[1]helpedfirm up the ideas about ergodic theory and is credited with the following early result in thefield. Theorem(Birkhoff).Consider the dynamical system(X,B,µ,T)where T is an ergodic map,and let f:X→R be a real-valued measurable function.Then for almost every x∈Xwe have that1limn→∞=µ(A),n+1i.e.,the average number of times that x shows up in A under T corresponds to the measure of A.Looking at(2)in another way we can use these sequences of averages to define an integral which in turn will define a measure.This will essentially be the idea when we construct an appropriate measure for our proof of Szemer´e di’s theorem.3Ergodic view of Szemer´e di’s theoremReturning to the problem offinding long arithmetic progressions,let us denote the shift map of integers by T,i.e.,T x→x+1,and let T m,m∈Z,denote repeated composition of the shift map.Then to say that B⊆Z contains an arithmetic progression of length k is equivalent to showing that for some n>0B∩T−n B∩T−2n B∩···∩T−(k−1)n B=0.To see this suppose that b lies in the intersection.Then since b∈T−n B that implies that b+n∈B,similarly b+2n,b+3n,...,b+(k−1)n∈B.The setb,b+n,b+2n,...,b+(k−1)nis the desired arithmetic progression.Examining our shift map we note that the only sets which are invariant under T−1is Z and ∅.At the same time,note that a set B has positive upper Banach density if and only if T−1B has positive upper Banach density.If we canfind some appropriate space and measure that takes these into account then we can use T as an ergodic map and then apply ergodic theory.4Constructing our measureWe start by lettingΩ={0,1}Z,i.e.,the space of all{0,1}-sequences.We can think of an element inΩas an indicator function for a subset of Z.Since{0,1}is compact then by Tychonoff’s Theorem we have thatΩis compact,further it has a corresponding metric. Also,let T:Ω→Ωbe the shift map,i.e.,Tω(n)=ω(n+1).For an elementξ∈Ωwe say that1occurs with positive upper Banach density if the set {n∈Z|ξ(n)=1}has positive upper Banach density.In particular,if B is a set which has positive upper Banach density then1occurs with positive upper Banach density inξ=1B. There turns out to be a correlation between sets with positive upper Banach density and T-invariant measures.Lemma(See[4],p.73).Letξ∈Ω,let X={T nξ|n∈Z}be its orbit closure with respect to the shift,and let A(1)={ω∈X:ω(0)=1}.Then if the symbol1occurs with positive upper Banach density then there exists a T-invariant measureµon X withµ(A(1))>0. Note by our assumptions we can assume that X is a compact metric space.Before beginning the proof it should be noted that this is an if and only if relationship,that is if there exists an invariant measureµon X withµ(A(1))>0then the symbol1occurs with positive upper Banach density.Since we do not need that for out argument we omit the proof.The interested reader is referred to the citation given.Proof.If the symbol1occurs inξwith positive upper Banach density then there exists a sequence of intervals{I k}achieving the limit as given in(1).For a given function f∈C(X) (the space of continuous real-valued functions),we can,by possibly passing to a subsequence, assume that1L(f):=limk→∞|I k| n∈I k f(T nξ)−f(T(n+1)ξ) =lim k→∞1Finally,we note thatµ(A(1))= X1A(1)dµ=L(1A(1))=lim k→∞1|I k|→ε>0.With this measure,Szemer´e di’s theorem would follow from the below result.Theorem(Multiple Recurrence Theorem(MRT),See[3],p.210,[4],p.77,151). Let(X,B,µ,T)be a measure preserving system and B∈B withµ(B)>0.Then for any integer k>1there exists n=0withµ(B∩T−n B∩T−2n B∩···∩T−(k−1)n B)>0.It should be noted that this is actually a special case of the MRT.In general we have that if T1,T2,...,T k−1are a set of commuting measure preserving transformations andµ(A)>0then for some n we haveµ(A∩T−n1A∩T−n2A∩···∩T nk−1A>0.The special case abovefollows by letting T i=T i.To see how Szemer´e di’s theorem follows from the MRT,suppose that S⊆Z has positive upper Banach density.Again let X⊂{0,1}Z be the orbit closure of1S∈{0,1}Z,and let B=B(1)={ω∈X|ω(0)=1}.Using the lemma there exists a measureµon X invariant under the shift and withµ(B)>0.Then by the MRT there will be some positive n and some pointωwith T inω∈B∩X for i=0,...,k−1.But this implies thatω(0)=ω(n)=···=ω((k−1)n)=1.Moreover sinceω∈X it is a limit of translates of1S so that for some b we have1S(b)=1S(b+n)=···=1S(b+(k−1)n)=1.So S contains the arithmetic progressions b,b+n,...,b+(k−1)n.5Poincar´e recurrence and two term arithmetic pro-gressionsWe will establish the MRT in stages.Thefirst step is the case when k=2.That is we need to argue that givenµ(A)>0then for some n thatµ(A∩T−n A)>0.This follows from the first result in ergodic theory,known as Poincar´e recurrence.Which says roughly that if B is a set withµ(B)>0then there is some x∈B and n so that T n x∈B.To argue this,assume thatµ(A∩T−n A)=0for all n≥1.Then for any i=j we must haveµ(T−i A∩T−j A)=0.So{T−iA}would form an essentially disjoint family of sets all of whom have the same nonzero measure,µ(T−i A)=µ(A)>0.This is impossible though becauseµ(X)<∞.6Roth’s theorem on three term arithmetic progres-sionsWe include the proof of the MRT in the special case that will allow us to do three term arithmetic progressions.The following pages are from[4,p.88-95].Along with the above discussion the proof is essentially self contained.We only need to introduce a few more terms.Definition.A point x is a generic point if it satisfies(2)for all functions f∈C(X). When T is a continuous map of a compact metric space to itself then almost every point of X is generic forµ.Definition(See[4],p.28).A subset S of an abelian group G is syndetic if there exists afinite set K⊂G such that for each element g∈G,there exists k∈K with gk∈S.In the case that G=Z this says that S is syndetic if there exists a positive integer N such that S∩[m,m+N]=∅for all integers m.References[1]G.D.Birkhoff,Dynamical Systems,Colloquim Publications IX,AMS,Providence,1927.[2]P.Erd˝o s and P.Tur´a n,“On some sequences of integers”,J.London Math.Soc.11(1936),261-264.[3]H.Furstenberg,“Ergodic behavior of diagonal measures and a theorem of Szemer´e di onarithmetic progressions”,J.D’Analse Math.31(1977),204-256.[4]H.Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory,Princeton University Press,Princeton,1981.[5]H.Furstenberg,2004Ordway lectures,/˜adams/Furstenberg[6]K.F.Roth,“Sur quelques ensembles d’entiers”,C.R.Acad.Sci.Paris234(1954),388-390.[7]E.Szemer´e di,“On sets of integers containing no four elements in arithmetic progres-sion”,Acta Math.Acad.Sci.Hunga.20(1969),89-104.[8]E.Szemer´e di,“On sets of integers containing on k elements in arithmetic progression”,Acta Arith.27(1975),199-245.[9]B.L.van der Waerden,“Beweis einer Baudetschen Vermutung”,Nieuw Arch.Wisk.15(1927),212-216.7。