Almost Arithmetic Progressions in "substantial’ subsets of Integers

Shobha Madan Currently at Indian Institute of Technology, Goa

October 17, 2020. Gonit Sora

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Infinitude of Prime Numbers

There are infinitely many primes

1. Euclid’s Proof (4th century BC):

Suppose not, i.e. there are only finitely many, say p1 < p2 < ... < pn. Consider the number

N = p1p2 ... pn + 1

If N is a prime and N > pn . If not, note that N is not divisible by any pk , k = 1, 2, ...n. By the Unique Factorization Theorem, there exists a prime p such that p|N, and p > pn.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers 2. Proof using Fermat Numbers

2n Theorem. The Fermat numbers Fn = 2 + 1 are pairwise coprime.

We can prove by induction that

n−1 Y Fn = Fk + 2 k=1

Now if p is a common divisor of Fk for some k < n and Fn, then p|(Fn − 2), so p|2, but all Fermat numbers are odd, and there are infinitely many.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers 3. Euler’s proof

If there are finitely many primes, p1, p2, ..., pn, we let

n Y 1 Pn = 1 − 1/pk k=1 Taking the truncated Taylor series expansion, for any m we have,

n Y 1 1 1 Pn ≥ (1 + + 2 + ... + m ) pk p p k=1 k k M X 1 = → ∞ j j=1

m m m where M = p1 p2 ...pn Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Density

So there are infinitely many primes. But how sparse or dense is the set of primes in N?

One measure of density of a subset A in N is to look at the proportion of elements of A asymptotically. More precisely, we define the upper asymptotic density of A by

#(A ∩ [1, N]) δA = lim sup N N

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers P δA > 0 implies n∈A 1/n diverges.

Suppose A ⊂ N, and δ = δA > 0. Let N0 = 1; we choose a 4Nk−1 sequence Nk such that Nk ≥ δ , and δ #(A ∩ [1, Nk )) ≥ 2 Nk for all k ∈ N. Then,

∞ X 1 X X 1 = n n n∈A k=1 n∈A∩[Nk−1,Nk ) ∞ X 1 ≥ (#(A ∩ [1, Nk )) − Nk−1) Nk k=1 ∞ ∞ X δ δ 1 X δ ≥ ( Nk − Nk ) = → ∞ 2 4 Nk 4 k=1 k=1

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers The converse of this result is not true. For the primes P, Euler proved in 1737 that

X 1 = ∞ p p∈P On the other hand, by the Prime Number theorem, it follows that the density δP = 0, since N π(N) = |{p prime : p ≤ N}| ∼ log N

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers ’Substantial’ sets

We now have two ways of saying that a set is "substantial’:

δA > 0

P 1 n∈A n

If a set is substantial (has enough elements!), then it is expected that some structures will exist in the set.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Erdös - Turán Conjecture 1

In 1936, Erdös and Turán conjectured in a paper ’On some sequences of Integers’, that every set of integers A with positive upper asymptotic density contains a k-term arithmetic progression for every k.

Erdös announced a prize of $1000 for a solution.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Paul Erdös Pàl Turan

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Klaus Roth

In 1953, K Roth proved the conjecture for the case k = 3, by using the Hardy-Littlewood Circle method (mentioned in his Fields Medal citation, 1958).

In 1969, Szemerédi proved the result for k = 4, and in 1975, the general case. His proof is combinatorial, and brilliant!

Szemerédi collected the $1000 Erdös prize!

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Klaus Roth Endre Szemerédi Fields Medal, 1958 Abel Prize 2012

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Other Proofs

There are now two other, different proofs of Szemerédi’s theorem: In 1977, Furstenberg and Katznelson gave an Ergodic Theory proof.

In 2001, Timothy Gowers gave a new proof using Fourier Analysis and Combinatorics. It is rather interesting that proofs of this theorem come from very diverse fields of Mathematics.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers H. Furstenberg Timothy Gowers Abel Prize 2020 Fields Medal 1998

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Erdös -Turan conjecture 2

Conjecture: Let a1 < a2 < ... be an infinite sequence of P integers satisfying 1/aj = ∞. Then for every k there are k a’s in an arithmetic progression.

In 1976 Erdös attached a prize of $2500 for resolution of this conjecture.

Later, in a paper Some of my favorite problems and results in 1996, he wrote: "I offer $5000 for a proof (or disproof) of this [problem]. Neither Szemerédi nor Furstenberg methods are able to settle this but perhaps the next century will see its resolution."

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Primes- a special case

Erdös: ’It is not known if there are infinitely many sets of three consecutive primes . S Chowla has demonstrated this without the restriction to consecutive primes. I was told that Chowla was anticipated by van der Corput’.

J. G. van der Corput. Uber Summen von Primzahlen und ´l Primzahlquadraten. Math. Ann., 116 (1939) 1-50.

CHOWLA (S . D .) . - There exists an infinity of 3-combinations of primes in A . P ., Proc . Lahore philos . Soc . 6 (1944) 15-16 .

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Green-Tao Theorem [2004] . The sequence of prime numbers contains arbitrarily long arithmetic progressions

The Green-Tao Theorem uses the very unique properties of the set of prime numbers and an amazing fusion of methods from analytic number theory and ergodic theory,

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Ben Green Terence Tao

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Most recently...

On July 7, 2020, Bloom and Sisak uploaded a paper on Arxiv, where they have proved the Erdös-Turan conjecture 2 for 3-term APs. It is generally believed that their proof is correct. The paper is under review by the referees.

Open: If divergence then APs of every finite length.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers "Almost" Arithmetic Progressions

Definition: A set of positive integers A is said to get arbitrarily close to arbitrarily long APs if, for all k and  > 0, there exists an arithmetic progression P of length k and gap size ∆ > 0 such that sup inf |p − a| ≤ ∆. p∈P a∈A

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers JM Fraser

Theorem (Fraser) + P∞ If A ⊂ Z , such that k=1 1/ak = ∞, then A is arbitrarily close to arbitrarily long APs.

The interest of this result lies in the fact that the Erdös-Turan conjecture, the genuine one, is still open.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Proof

We define the upper logarithmic density by

log |A ∩ [m + 1, m + n]| δlog = lim sup sup n→∞ m≥0 log n

Lemma (1)

If A = {ak } is a subset of integers such that

∞ X 1 = ∞, ak k=1

Then δlog = 1.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Proof of Lemma 1

Suppose the upper logarithmic density δlog < 1. Let ρ be such that δlog < ρ < 1. Then there exists an integer N such that for all n ≥ N and for all m,

log #(A ∩ [m + 1, m + n)) ≤ ρ log n or #(A ∩ [m + 1, m + n)) ≤ nρ

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers n n+1 For integers n ≥ N, let An = A ∩ [2 , 2 ), then

ρn #(An) ≤ 2 .

Then,

∞ X X X 1/ak = 1/ak N k≥2 n=N {k:ak ∈An} ∞ ∞ X −n X (ρ−1)n ≤ #(An)2 ≤ 2 < ∞ n=N n=N P This is a contradiction, since we assumed k 1/ak = ∞.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Lemma 2

Lemma (2)

If A = {ak } is a subset of integers such that its upper logarithmic density δlog = 1, then A is arbitrarily close to arbitrarily long APs.

Proof: Suppose not, then there exists some  > 0 and a k such that given any AP P of length k and gap size ∆,

sup inf |p − a| > ∆. p∈P a∈A

We may assume that 1/2 = M, a large integer. We fix an interval J = [m + 1, m + n] Then |J| = n − 1.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Subdivide this interval into kM equal intervals of length |J|/kM, and label them from left to right as 1, 2, 3, ..., kM.

Now form congruence classes modulo M on these labels and associated intervals. Then the centers of the intervals in the same equivalence class form an arithmetic progression of length k and gap size |J|/k

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers At least one interval from each equivalence class does not intersect A, since we have assumed

sup inf |p − a| > ∆. p∈P a∈A so A ∩ J is contained in the union of (k − 1)M intervals of length |J|/kM.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Iterate this step on each of the subintervals which intersect A. After N applications, we find that A ∩ J is contained in the union of at most ((k − 1)M)N intervals of 1 N length |J|( kM ) . Choose N to be the smallest integer such that 1 |J|( )N < 1 kM Then each subinterval at the Nth step contains at most one element of A.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers Finally we get

#A ∩ [m + 1, m + n] ≤ ((k − 1)M)N log(n−1) +1 ≤ ((k − 1)M) log(kM) log(k−1)M ≤ ((k − 1)M)n log(kM) for the second inequality, by our choice of N we have 1 N−1 |J|( kM ) > 1, so that log(n − 1) N − 1 ≤ log(kM)

The last inequality is just that αlog β = βlog α

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers We have got the inequality

log((k−1)M) #A ∩ [m + 1, m + n] ≤ ((k − 1)M)n log(kM) which implies log(#A ∩ [m + 1, m + n) log((k − 1)M) log((k − 1)M) ≤ + log n log n log(kM) log((k − 1)M) → < 1 log(kM) i.e. δlog < 1.

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers THANK YOU

Shobha Madan Almost Arithmetic Progressions in "substantial’ subsets of Integers