Arithmetic progressions in primes

Transcription

1 Arithmetic progressions in primes Alex Gorodnik CalTech Lake Arrowhead, September 2006 Introduction Theorem (Green,Tao) Let A P := {primes} such that Then for any k N, for some a, d N. lim sup N A [1, N] P [1, N] > 0. {a, a + d,..., a + (k 1)d} A

2 Previous results Van der Corput showed in (k = 3)-case that #{a, d [1, N] : a, a + d, a + 2d P} γ 3 N 2 (log N) 3 Conjecture (Hardy,Littlewood) #{a, d [1, N] : a, a + d,..., a + (k 1)d P} γ k for explicit γ k > 0. Compare with a random subset of [1, N] of density 1 log N. Green-Tao proof gives a lower estimate γ k N 2 (log N) k for small γ k > 0. N 2 (log N) k Conjecture (Erdos,Turan) Every set A N such that a A 1 a = contains arbitrary long arithmetic progressions.

3 Szemerédi Theorem Theorem (Szemerédi) Let A N such that Then for any k N, for some a, d N. lim sup N 1 A [1, N] > 0. N {a, a + d,..., a + (k 1)d} A Note that P [1, N] N log N as N. Almost primes P R := {numbers with all prime factors R}. Then for R = N α with small α > 0, P R [1, N] contains arbitrary long AP as N. lim sup N P [1, N] P R [1, N] > 0. Naive Strategy: Prove an analog of Szemerédi theorem for subsets of almost primes (relative Szemerédi theorem).

4 Effective Semerédi theorem Theorem Given k 3 and δ > 0, there exists N 0 = N 0 (k, δ) > 0 such that for every N > N 0 and every A [1, N] with A > δn, the set A contains an arithmetic progression of length k. Upper estimate (Bourgain): N 0 (3, δ) exp(δ 2 log(1/δ)). Upper estimate (Gowers): N 0 (k, δ) exp(e δ c k ). Lower estimate (Rankin): N 0 (k, δ) exp(log(1/δ) d k ). Expected: N 0 (k, δ) exp(c k δ 1 ) AP in primes. Szemerédi Theorem (other formulation) Z N = the field of residues mod N, E(f (x) x A) = 1 f (x) A x A Theorem (Szemerédi) Given k 3 and δ > 0, there exists c = c(k, δ) > 0 such that for a function f : Z N R satisfying we have 0 f 1 and E(f ) δ, E(f (x)f (x + r) f (x + (k 1)r) x, r Z N ) c for sufficiently large N.

5 Relative Szemerédi Theorem Theorem (Green, Tao) Fix k 3 and δ > 0, ν : Z N R + be a k-pseudorandom function with E(ν) = 1 + o(1). Then for any function f : Z N R + satisfying we have 0 f ν and E(f ) δ, E(f (x)f (x + r) f (x + (k 1)r) x, r Z N ) c(k, δ) o k,δ (1) as N. Pseudorandom function is defined to satisfy linear form condition, correlation condition. Linear form condition A function ν : Z N R + is called a measure if E(ν) = 1 + o(1). A measure ν satisfies (m 0, t 0, L 0 )-linear form condition if for any m m 0, t t 0, and linear forms ψ i (x) = t L ij x j + b i, i = 1,..., m, j=1 where L ij are rationals (assume N is prime) with the numerator/denominator bounded by L 0 and the t-tuples (L ij ; j = 1,..., t) are not zero and not rational multiples of each other and b Z N, we have E(ν(ψ 1 (x)) ν(ψ m (x)) x Z t N ) = 1 + o m 0,t 0,L 0 (1).

6 Hardy-Littlewood conjecture Linear form condition roughly says that the events ψ j (x) is almost prime are essentially independent. This is the Hardy-Littlewood prime tuples conjecture. Define Λ(n) = { log p for n = p k, 0 otherwise, Note that E(Λ) = 1 + o(1). Conjecture (Hardy-Littlewood) Assuming that L ij and b i are positive, E(Λ(ψ 1 (x)) Λ(ψ m (x)) x Z t N ) = α + o m 0,t 0,L 0 (1) for explicit α = α(ψ 1,..., ψ m ) > 0. Correlation condition A measure ν satisfies m 0 -correlation condition if for every m = 2,..., m 0, there exists a function τ = τ m : Z N R + such that E(τ q ) = O m,q (1) for all q 1 and E(ν(x + h 1 ) ν(x + h m ) x Z N ) 1 i<j m τ(h i h j ) for all h 1,..., h m Z N. A measure ν is called k-pseudorandom if it satisfies (k2 k 1, 3k 4, k)-linear form condition and 2 k 1 -correlation condition.

7 Relative Szemerédi Theorem Theorem (Green, Tao) Fix k 3 and δ > 0, and let ν be a k-pseudorandom measure. Then for any function f : Z N R + satisfying we have 0 f ν and E(f ) δ, E(f (x)f (x + r) f (x + (k 1)r) x, r Z N ) c(k, δ) o k,δ (1) as N. Finitary ergodic theory (X, µ, T ) (Z N, uniform measure, x x + 1). We are interested in averages as N. E(f 0 (x)t i f 1 (x) T (k 1)i f k 1 (x) x, i Z N ) For a probability measure preserving system (X, µ, T ), the Koopman von-neumann decomposition is L 2 (X ) = {weakly mixing} {compact (almost periodic)}. is used in the proof of Furstenberg s multiple recurrence. A finitary analog of Koopman von-neumann decomposition is crucial in the proof of relative Szemerédi theorem.

8 Sketch of the proof (step 1) The crucial step is to show: Koopman von-neumann decomposition : f = f U + f U + E with the error term E satisfying E(E) = o(1), and E 0, f U + f U 0. Then E(f (x) f (x + (k 1)r) x, r Z N ) E((f U + f U )(x) (f U + f U )(x + (k 1)r) x, r Z N ). The function f U is uniform (weakly mixing) component of f : E(f U ) = o(1), E(f 0 (x)f 1 (x + r) f k 1 (x + (k 1)r) x, r Z N ) is small, where each f i is either f U or f U, and f i f U for some i. Sketch of the proof (step 2) It suffices to prove a lower estimate for E(f U (x)f U (x + r) f U (x + (k 1)r) x, r Z N )?. The antiuniform (almost periodic) component f U satisfies 0 f U 1 + o(1). Hence, the lower estimate follows from the classical Szemerédi theorem.

9 Gowers uniformity norms These norms are used to controle multiple averages. Lemma (Van der Corput) For f : Z N R, E(f ) 2 = E(E(f T h f ) h Z N ). Gowers uniformity norms are defined inductively: or equivalently, f U d f U 1 = E(f ), f U k = E = E( f T h f 2k 1 U k 1 h Z N ) 1/2k. ω {0,1} d f (x + ωh) x Z N, h Z d N 1/2d. Gowers uniformity norms For a family of functions f ω, ω {0, 1} d, we define d-dimensional Gowers inner product: (f ω ) ω {0,1} d U d = E f ω (x + ωh) x Z N, h Z d N. ω {0,1} d Then we have Gowers Cauchy-Schwartz inequality: (f ω ) ω {0,1} d U d f ω U d. ω {0,1} d

10 Generalized von Neumann theorem Theorem For a k-pseudorandom measure ν : Z N R +, λ 0,..., λ k 1 Z, λ i λ j, functions f 0,..., f k 1 : Z N R such that f i ν + 1, we have E ( k 1 i=0 ) ( f i (x + λ i h) x, h Z N = O min 0 i k 1 f i U k 1 ) + o(1).