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2015 Difference set of primes and related problems Wen Huang Difference set of primes and related problems 1.Background 2.A conjecture 3. Higher order Bohr Wen Huang problem 1).Furstenberg Joint work with Xiaosheng Wu correspondence principle 2). Multiple ergodic average 3).A result Dept of Math., SCU+USTC 4. Main result April, 2015

Outline µ Difference set of primes and related problems Wen Huang Background 1 1.Background 2.A conjecture A conjecture 2 3. Higher order Bohr problem 1).Furstenberg correspondence principle Bohr problem 3 2). Multiple ergodic average 3).A result 4. Main result Main result 4

1. Background Difference set of primes and Given a subset S = { s 1 < s 2 < · · · } of natural numbers related problems N = { 1 , 2 , · · · } , we may consider it as a ”big set” as Wen Huang following: 1.Background S is called syndetic, if there is L ∈ N such that 1 2.A conjecture { i , i + 1 , · · · , i + L − 1 } ∩ S � = ∅ for any i ∈ { 1 , 2 , · · · } ; 3. Higher order Bohr S has positive upper Banach density, if 2 problem 1).Furstenberg correspondence principle 1 BD ∗ ( S ) = lim sup 2). Multiple ergodic N | S ∩ { M , M + 1 , · · · , M + N − 1 }| sup average 3).A result N → + ∞ M ∈ N 4. Main result is larger than zero. os condition, if � ∞ 1 s i = + ∞ . For S satisfies Erd¨ 3 i = 1 example: Prime numbers P .

1. Backgroud Difference set of primes and related problems A basic qustion: µ Does a ”big subset” of natural numbers Wen Huang contain a ”good” linear structure? eg µ Are there any arbitrarily long arithmetic progressions in 1.Background Prime numbers set P ? 2.A conjecture 3. Higher order Bohr problem In 1939 van der Corput prove the case k = 3 for prime 1).Furstenberg correspondence set P . principle 2). Multiple ergodic average edi Theorem(1975 § Cominatorial method): A Szemer´ 3).A result subset of natural numbers with positive upper Banach 4. Main result density contains arithmetic progressions of arbitrary finite length.

1. Background Difference set of primes and related problems Wen Huang edi proved that for any δ > 0 and k ≥ 3 , In fact, Szemer´ 1.Background there is a minimal natural number N ( δ, k ) such that If 2.A conjecture N ≥ N ( δ, k ) and A ⊆ { 1 , 2 , · · · , N } with | A | ≥ δ N , then A 3. Higher order Bohr contains an arithmetic progression of length k . problem 1).Furstenberg (In 1953, Roth proved the case k=3.) correspondence principle 2). Multiple ergodic In 1977, Furstenberg gave a ergodic theory’s proof for average 3).A result Szemer´ edi Theorem. 4. Main result

1. Background Difference set of primes and related problems Gowers(1998,2001) gave a new proof for Szemer´ edi Wen Huang Theorem, in particular he proved 1.Background N ( δ, k ) ≤ 2 2 δ − ck , c k = 2 2 k + 9 . 2.A conjecture 3. Higher order Bohr For k = 3 , Roth Theorem(1953), Heath-Brown(1987), problem 1).Furstenberg Szemer´ edi(1990). correspondence principle 2). Multiple ergodic average Bourgain(1999) proved: 3).A result 4. Main result N ( δ, 3 ) ≤ 2 C δ − 2 log ( 1 δ ) . Bourgain(2003), Green(2005) ect..

1. Background Difference set of primes and related problems Wen Huang edi . s theorem In 2008, Green and Tao proved (Szemer´ in the primes). Let A be any subset of the prime 1.Background numbers of positive relative upper density: if 2.A conjecture | A ∩{ 1 , 2 , ··· , N }| 3. Higher lim sup N → + ∞ | P ∩{ 1 , 2 , ··· , N }| > 0 , then A contains infinitely order Bohr problem many arithmetic progressions of length k for all k . 1).Furstenberg correspondence Green,Tao,Ziegler...... principle 2). Multiple ergodic average os conjecture µ Suppose that A satisfies Erd¨ Erd¨ os 3).A result condition, then A contains arbitrarily long arithmetic 4. Main result progressions.

1. Background: our question Difference set of primes and related Now our question is µ problems Given k ≥ 1 . If S is a / big set 0 of natural numbers, then Wen Huang all common differences n of arithmetic progressions 1.Background a , a + n , a + 2 n , · · · , a + kn with length k + 1 appeared in the 2.A conjecture subset S forms a set 3. Higher order Bohr problem C k ( S ) := { n ∈ N : S ∩ ( S − n ) ∩ ( S − 2 n ) ∩ · · · ∩ ( S − kn ) � = ∅} . 1).Furstenberg correspondence principle 2). Multiple ergodic How about the structure of the set C k ( S ) ? average 3).A result For example µ For prime numbers P , Is C k ( P ) syndetic? 4. Main result Recall that C k ( P ) is the set formed by all common differences of arithmetic progressions with length k + 1 appeared in P .

2. A conjecture Difference set of primes and related problems Wen Huang In fact, we have the following cojecture µ 1.Background 2.A conjecture Conjecture 3. Higher Given k ≥ 1 . If S is a / big set 0 of natural numbers, then order Bohr problem 1).Furstenberg correspondence C k ( S ) := { n ∈ N : S ∩ ( S − n ) ∩ ( S − 2 n ) ∩ · · · ∩ ( S − kn ) � = ∅} principle 2). Multiple ergodic average 3).A result is an ”almost” Nil k Bohr 0 set? 4. Main result

2. A conjecture Difference set of primes and related problems Where a subset A of natural numbers is called Nil k Bohr 0 Wen Huang set, if there exists a k -step nilsystem ( X , T ) , x 0 ∈ X and an open neighborhood U of x 0 such that 1.Background 2.A conjecture A ⊇ { n ∈ N : T n x 0 ∈ U } . 3. Higher order Bohr problem 1).Furstenberg correspondence principle Remark 2). Multiple ergodic average 3).A result Nil k Bohr 0 set is also called Bohr 0 set; Bohr 0 set is a classic 4. Main result notion; Nil k Bohr 0 sets were introduced by Host and Kra recently; Each Nil k Bohr 0 set is syndetic.

k -nilsystem Difference set of primes and related problems Definition Wen Huang Let G be a k -step nilpotent Lie group and Γ a discrete 1.Background cocompact subgroup of G . The compact manifold X = G / Γ 2.A conjecture is called a k -step nilmanifold. The group G acts on X by left 3. Higher order Bohr translations and we write this action as ( g , x ) �→ gx . The problem Haar measure µ of X is the unique probability measure on X 1).Furstenberg correspondence principle invariant under this action. 2). Multiple ergodic average 3).A result Let τ ∈ G and T be the transformation x �→ τ x of X , i.e the 4. Main result rotation induced by τ ∈ G . Then ( X , T , µ ) is called a k -step nilsystem.

k -nilsystem Difference set of primes and related problems Remark Wen Huang Rotation on torus is 1 -step nilsystem. Katznelson(2001) proved: A subset A of natural numbers is Bohr 0 set if and 1.Background 2.A conjecture only if ∃ a 1 , a 2 , · · · , a m ∈ R and ǫ > 0 such that 3. Higher order Bohr m problem � A ⊇ { n ∈ N : na i ( mod Z ) ∈ ( − ǫ, ǫ ) } . 1).Furstenberg correspondence principle i = 1 2). Multiple ergodic average 3).A result 4. Main result In later, one of Huang+Shao+Ye results is to give a similar characterization of Nil k Bohr 0 set by generalized polynomials of degree k .

3.Reasons of Conjecture µ Higher order Bohr problem Difference set of primes and related problems • In 1968, W.Veech proved µ Wen Huang Theorem (Veech) 1.Background 2.A conjecture Let S ⊆ N be a syndetic set. Then 3. Higher order Bohr S − S := { s 1 − s 2 : s 1 , s 2 ∈ S , s 1 > s 2 } problem 1).Furstenberg correspondence principle (That is, the common differences of arithmetic progressions 2). Multiple ergodic average with length 2 appeared in S ) is an / almost 0 Bohr 0 set, 3).A result More precise, there exists a set M with BD ∗ ( M ) = 0 such 4. Main result that ( S − S ) ∪ M is a Bohr 0 set.

Bohr Problem Difference set of primes and related problems Wen Huang 1.Background Problem (Bohr problem) 2.A conjecture Let S ⊆ N be a syndetic set. Is S − S a Bohr 0 set? 3. Higher order Bohr problem Furstenberg(1981), Glasner(1998,2004), Weiss(2000), 1).Furstenberg correspondence principle Kantznelson(2001), Bergelson+ Furstenberg+Weiss(2006), 2). Multiple ergodic average Host+Kra(2011), Huang+Ye(2002,2011) � 3).A result 4. Main result

Furstenberg correspondence principle Difference set of primes and related problems Wen Huang Problem (Higher order problem) 1.Background Let S ⊆ N be a syndetic set. How about the structure of the 2.A conjecture set 3. Higher order Bohr problem { n ∈ N : S ∩ ( S − n ) ∩ ( S − 2 n ) ∩ · · · ∩ ( S − kn ) � = ∅} 1).Furstenberg correspondence principle 2). Multiple ergodic average Similarly, we may consider the case that S has positive 3).A result upper Banach density? 4. Main result