View
65
Download
2
Category
Tags:
Preview:
DESCRIPTION
On set of integers containing no k elements in arithmetic progression. E. Szemerédi Acta Arithmetica, 1975 Shuchi Chawla. The Problem and some History. [van der Waerden, 1926] Let N = S 1 [ S 2 . Then either S 1 or S 2 contains arbitrarily long APs. - PowerPoint PPT Presentation
Citation preview
On set of integers containing no k elements in arithmetic progression
E. SzemerédiActa Arithmetica, 1975
Shuchi Chawla
Shuchi Chawla, Computer Science 2
The Problem and some History
[van der Waerden, 1926] Let N = S1[S2. Then either S1 or S2 contains arbitrarily long APs.
Erdós and Türan [1936] defined rk(n) – the largest l for which an l -element sequence 2 {1,…,n} does not contain a k-term AP.
How big can rk(n) be? How does it grow with n?
Shuchi Chawla, Computer Science 3
The Problem and some History
We show [Sz., 1975] that:limn!1 rk(n)/n = 0
This was the motivation behind the Regularity Lemma
Shuchi Chawla, Computer Science 4
The Problem and some History
We show [Sz., 1975] that:limn!1 rk(n)/n = 0
In other words:For every and k, 9 N(k,), such that for all n¸N(k,), rk(n)· n
Shuchi Chawla, Computer Science 5
The Problem and some History
We show [Sz., 1975] that:limn!1 rk(n)/n = 0
In other words:For every and k, 9 N(k,), such that for all n¸N(k,), rk(n)· n
Equivalently,For every set R with +ve upper density, R contains arbitrarily long APs
Shuchi Chawla, Computer Science 6
The Problem and some History
We show [Sz., 1975] that:limn!1 rk(n)/n = 0
In other words:For every and k, 9 N(k,), such that for all n¸N(k,), rk(n)· n
Equivalently,For every set R with limn!1|RÅ{1,…,n}|/n > 0, R contains arbitrarily long APs
Not obvious!!
Shuchi Chawla, Computer Science 7
From 1 sets to a bound on rk(n)
Assume 9 n1·n2·… and Riµ[0,ni) with |Ri|>n and Ri contains no k-term AP
Let {n’i} be a subseq with n’i+1¸3n’i and di=j<in’j R’ = [i (Rn’i+di) Note that each R gets mapped to
a disjoint set R’ has +ve U.D. ) it contains a sequence of 3k-
terms. Say A = {a+di | 0·i·3k} Let Rn’l be the last set in this. Then either this or the
second last one must contain k terms.
Shuchi Chawla, Computer Science 8
The Plan
Regularity Lemma
Definitions
Main Proof
Shuchi Chawla, Computer Science 9
A “few” definitions
Configurations of order m B(l1,…,lm) X2B(l1,…,lm), then, X=[iXi Xi2B(l1,…,lm-1) eg. (1,2, 5,6, 9,10)2 B(2,3)
t1,…,tm – numbers arising from reglem Saturated and Perfect configurations
S(l1,…,lm)½B(l1,lm) sm(X) = #i : Xi2S(l1,…,lm-1) gm(l) = max {sm(X) : X2B(t1,…,tm-1,l)} – rate of convergence of g(l)/l ; – distance from
pm, fm, m and m defined analogously
Shuchi Chawla, Computer Science 10
Saturated Sets
S(;) = B(;) = {{n} : n2N}
S(t1,…,tm) = {X : sm(X)¸( m-m )tm and pm(X)¸( m-(..) )tm}
A large fraction of Xi are saturated and a large fraction are perfect
Shuchi Chawla, Computer Science 11
R-equivalence and Perfect Sets
X and Y are R-equivalent if: for corresponding elements x2 X and y2 Y, x2 R , y2 R
P(t1,…,tm) is the “largest” equivalence class
P(;) = {{n} : n2R}
Shuchi Chawla, Computer Science 12
More Definitions
C(t1,…,tm,l) = {X2B(t1,…,tm,l) : sm+1(X)=l} i.e., all Xi are saturated
Fact: For appropriate choice of tis, S, P and C are non empty.
Di(t1,…,tm,K) = {X2C : all j<i have Xj2P}
Main Theorem: Dk-1(k) is non empty.Proof by induction on i that for fixed k and any m, Di(t1,…,tm,K);
Shuchi Chawla, Computer Science 13
Further More Definitions
E(t,K) = all K-term APs with each term·t Fact: Given X2 B(t1,…,tm,K),
{ji} 2 E(tm,K) , [i<KXi,ji2 B(t1,…,tm,K)
E(t,K,j,i) = all APs in E(t,K) with j as the ith element e(t,K,j,i) = |E(t,K,j,i)|
Fact: e(t,K,j,i)·t and if t/4·j·3t/4, e(t,K,j,i)¸ t/K2
Shuchi Chawla, Computer Science 14
More Definitions (Last)
F(X,j,i,s) = all APs in E(tm,K,j,s) such that Xi’,j’2Di(t1,…,tm,K) f(X,j,i,s) = |F(X,j,i,s)|
Gi(t1,…,tm,K) = those X in C, such that for every s:
f(X,j,i,s) · 2imitm fails for · 2im
K(1-m)tm indices j, j · tm, and, f(X,j,i,s) ¸ 1/K2 ½im
itm fails for · 2imK(1- m)tm
indices j, tm/4 · j · 3tm/4
Shuchi Chawla, Computer Science 15
Finally… the Graph!
This sequence is in F(X,j,i,s) and
ji=j’or
All these sets are perfect
B A
X0,0 … Xi,0 … Xs,0 …X0,1 … Xi,1 … Xs,1 …
X0,tm-1 … Xi,tm-1 … Xs,tm-1 …
Xi,j’
Xs,j………
…
I(X,i,s) :
Shuchi Chawla, Computer Science 16
The Proof: Part 1
X is well-saturated if for all s, | pm(Xi,C,(s)) - m|C,(s)| |· |C,(s)| (whenever C,(s) is large enough)
Lemma 4: If X2Gi is well saturated, then X2Gi+1
Proof uses regularity
Shuchi Chawla, Computer Science 17
The Proof: Part 2
Lemma 5: Suppose for all <l, X()2 Gi(t1,…,tm,K) and X()
j and X()j are R-equivalent for all ,,j<i,
[<l X()i 2 C, then one of them is in Gi+1
Proof by contradiction: Show that one of them has to be well saturated.
Shuchi Chawla, Computer Science 18
The Proof: Part 3
Lemma 6: X() are as before. Then, there exists a sequence of lm APs in E(tm,K) such that their ith elements form an AP and for each X and for each AP, [i’<K Xi’j’ 2 Di
Proof: Consider any X. Define Z to be the set of indices j such that f(j,i,i) is non empty. Show that this set contains an AP of length lm. Since Xs are R-equivalent, the conditions will hold for every X.
Shuchi Chawla, Computer Science 19
One more definition…
If Y2 B(t1,…,tm,K) and X2 B(t1,…,tm’,K), and, Yi is a subconfig of Xi, we write Y|X
If Y|X and Y’|X’, we say that the position of Y in X is the same as that of Y’ in X’ if for each i, Yi is the jith subconfig of Xi and same for Y’i.
Shuchi Chawla, Computer Science 20
The Proof: Part 4
Fact 12: For every m, and m’¸ h(m,i) and X()2 Di(t1,…,tm’,K), there exist Y()2 Gi(t1,…,tm,K) such that Y()|X() and the position of the Ys are the same in the Xs
Proof: induction on i
Shuchi Chawla, Computer Science 21
The Proof: Part 5
Theorem: 8 m,i,K Di(t1,…,tm,K);
Proof: induction on i
Shuchi Chawla, Computer Science 22
Summary of results
Lem 4: Gi & well-sat ) Gi+1
Lem 5: Gi & R-equiv for j· i ) Gi+1
Lem 6: Gi & R-equiv ) Di
Fact 12: Di ) Gi
Theorem: non empty Di ) non empty Di+1
Shuchi Chawla, Computer Science 23
Concluding Remarks
Paper by Tim GrowersA new proof of Szemerédi’s
Theoremhttp://www.dpmms.cam.ac.uk/~wtg10/papers.html
(129 pages!!!)
Recommended