Sumsets via Spectral Properties of Dynamical Systems John Griesmer University of Denver Ergodic Theory with Connections to Arithmetic University of Crete June 4, 2013 John Griesmer (U Denver) Sumsets via Spectral Properties June 4 1 / 19
These slides are mostly based on the article Sumsets of dense sets and sparse sets , Israel Journal of Mathematics, Volume 90, issue 1, pp 229-252, doi 10.1007/s11856-012-0008-1, preprint at arXiv:0911.2278 John Griesmer (U Denver) Sumsets via Spectral Properties June 4 2 / 19
Definitions and notation The upper Banach density of E ⊂ Z is | E ∩ I| d ∗ ( E ) := lim sup |I| k →∞ |I| = k I an interval If F ⊂ Z is finite, we say the finite configuration F appears in E if F + t ⊂ E for some t ∈ Z . Observation: the upper Banach density of E depends only on the finite configurations appearing in E . John Griesmer (U Denver) Sumsets via Spectral Properties June 4 3 / 19
Szemer´ edi’s Theorem Theorem (Szemer´ edi) If E ⊂ Z has d ∗ ( E ) > 0 , then E contains arithmetic progressions of every finite length. Equivalent to Theorem (Furstenberg) If ( X , µ, T ) is a probability measure preserving system and D ⊂ X has µ ( D ) > 0 , then for all k, there exists n > 0 such that µ ( D ∩ T − n D ∩ T − 2 n D ∩ · · · ∩ T − ( k − 1) n D ) > 0 Many generalizations and refinements followed. John Griesmer (U Denver) Sumsets via Spectral Properties June 4 4 / 19
Refinements of Szemer´ edi’s Theorem A prototypical refinement: Theorem (Bergelson and Leibman) If d ∗ ( E ) > 0 and k ∈ N , then E contains a set of the form { a , a + n 2 , a + 2 n 2 , · · · , a + ( k − 1) n 2 } . Continuing problem: what kinds finite configurations necessarily appear in sets E having d ∗ ( E ) > 0? Many positive results, due to: Austin, Balister, Bergelson, Chu, Conze, Frantzikinakis, Furstenberg, Green, Host, Katznelson, Kra, Leibman, Lesigne, McCutcheon, Tao, Wierdl, Ziegler, and Zorin-Kranich, in various combinations. John Griesmer (U Denver) Sumsets via Spectral Properties June 4 5 / 19
Sumsets We now consider sets of the form A + B := { a + b : a ∈ A , b ∈ B } , where A , B ⊂ Z . (The sumset of A and B ) What finite configurations must appear in A + B ? When d ∗ ( A ) > 0 and d ∗ ( B ) > 0, there is a precise answer. Definition A set S ⊂ Z is a Bohr set if there is 1 A compact abelian group Z 2 An open set U ⊂ Z 3 An α in Z such that { n α } n ∈ Z is dense in Z such that S ⊃ { n : n α ∈ U } . A set E is piecewise Bohr (PW-Bohr) if there is a Bohr set S such that E contains all the finite configurations contained in S ; meaning for all finite F ⊂ S , F + t ⊂ E for some t . John Griesmer (U Denver) Sumsets via Spectral Properties June 4 6 / 19
examples of Bohr and PW-Bohr sets Definition A set S ⊂ Z is a Bohr set if there is 1 A compact abelian group Z 2 An open set U ⊂ Z 3 An α in Z such that { n α } n ∈ Z is dense in Z such that S ⊃ { n : n α ∈ U } . A set E is piecewise Bohr (PW-Bohr) if there is a Bohr set S such that E contains all the finite configurations contained in S ; meaning for all finite F ⊂ S , F + t ⊂ E for some t . √ S:= { n : n 2 mod 1 ∈ (0 , 1 / 4) } is a Bohr set with Z = R / Z , √ U = (0 , 1 / 4), and α = 2. E := S ∩ � ∞ n =1 [ n 2 , n 2 + n ] is piecewise Bohr. John Griesmer (U Denver) Sumsets via Spectral Properties June 4 7 / 19
Sumsets with dense summands are PW-Bohr Theorem (Bergelson, Furstenberg, Weiss, 2006) If d ∗ ( A ) > 0 and d ∗ ( B ) > 0 then A + B is PW-Bohr. Generalized to amenable groups by Bergelson, Beiglb¨ ock, and Fish. Strengthens Jin’s Theorem, which says that such A + B are piecewise syndetic. Jin’s Theorem was generalized to Z d by Jin and Keisler. Ultrafilter proof due to Beiglb¨ ock. Very elegant, very general approach forthcoming due to Bj¨ orklund and Fish. Theorem ( Beiglb¨ ock, Bergelson, Fish, 2009) If S is PW-Bohr, then S ⊃ A + B where d ∗ ( A ) > 0 and d ∗ ( B ) > 0 . John Griesmer (U Denver) Sumsets via Spectral Properties June 4 8 / 19
When d ∗ ( B ) > 0 and d ∗ ( A ) = 0 Let S = {⌊ n 5 / 2 ⌋ : n ∈ Z } . Define the upper density relative to S : | A ∩ S ∩ [1 , . . . N ] | d S ( A ) := lim sup | S ∩ [1 , . . . N ] | N →∞ Main Theorem (G., 2009) If d S ( A ) > 0 and d ∗ ( B ) > 0 , then A + B is piecewise Bohr. John Griesmer (U Denver) Sumsets via Spectral Properties June 4 9 / 19
Outline of the proof, I I. Translate to a statement about measure preserving systems: standard Furstenberg correspondence principle argument. We omit the precise statement of the correspondence principle. Let ( X , µ, T ) be an ergodic probability measure preserving system. Fix A = { a 1 , a 2 , a 3 , . . . } ⊂ Z with d S ( A ) > 0 and D ⊂ X with µ ( D ) > 0. � T a D . (Suggested to me by M. Bj¨ Study unions orklund and A. Fish.) a ∈ A Since f := 1 D is supported on D , the averages N � F N := 1 f ◦ T − a n N n =1 are supported on � a ∈ A T a D . John Griesmer (U Denver) Sumsets via Spectral Properties June 4 10 / 19
Outline of proof, II II. Reduce from general measure preserving systems to group rotations. Key fact: for all θ ∈ (0 , 1) N � 1 exp(2 π i θ ⌊ n 5 / 2 ⌋ ) = 0 lim N N →∞ n =1 (not just irrational θ ) In the limit of N � F N := 1 f ◦ T − a n N n =1 only the eigenfunctions of T matter. We need only consider ergodic group rotations. John Griesmer (U Denver) Sumsets via Spectral Properties June 4 11 / 19
Outline of proof, III N � 1 f ◦ T − a n when ( X , µ, T ) = ( Z , m , R ), where III. Identify lim N N →∞ n =1 • Z is a compact abelian group with Haar measure m • R : Z → Z is given by Rz = z + α . Think of the average as a convolution of measures ν N ∗ ( f dm ) N � where ν N = 1 δ a n α , N n =1 Now ν := lim N →∞ ν N ≪ m , and ν ( Z ) = d S ( A ) (use equidistribution) So ν N ∗ ( f dm ) → ν ∗ f dm , convolution of two functions in L 2 ( m ), continuous. John Griesmer (U Denver) Sumsets via Spectral Properties June 4 12 / 19
Outline of proof, IV IV. So ν N ∗ ( f dm ) → ν ∗ f dm , convolution of two functions in L 2 ( m ). We have shown N � 1 f ◦ T − a n F := lim N N →∞ n =1 is continuous (actually F = a continuous function m -a.e.) So � a ∈ A T a D contains an open set (up to m -measure 0) The correspondence principle turns this into the desired conclusion. � John Griesmer (U Denver) Sumsets via Spectral Properties June 4 13 / 19
Perfect squares vs ⌊ n 5 / 2 ⌋ . You can replace ⌊ n 5 / 2 ⌋ with any increasing sequence c n having the equidistribution property and get the same conclusion. N � 1 Equidistribution of c n means lim exp(2 π i θ c n ) = 0 for all θ ∈ (0 , 1) . N n →∞ n =1 What about Q = { n 2 : n ∈ N } ? Not equidistributed – take θ = 1 4 . Is Q + B PW-Bohr whenever d ∗ ( B ) > 0? No: Example There is a set B ⊂ Z having d ∗ ( B ) > 1 − ε such that Q + B is not PW-Bohr (not even piecewise syndetic). John Griesmer (U Denver) Sumsets via Spectral Properties June 4 14 / 19
Example There is a set B ⊂ Z having d ∗ ( B ) > 1 − ε such that Q + B is not PW-Bohr (not even piecewise syndetic). Construction: Let Z = Z / 2 Z × Z / 3 Z × Z / 5 Z × · · · , the direct product of the cyclic groups of prime order. Let α = (1 , 1 , 1 , 1 , . . . ), so that n α is dense in Z . Now � Q := { n 2 α : n ∈ Z } has Haar measure 0. There is a compact K ⊂ Z such that m ( K ) = 1 − ε and � Q + K has empty interior. Use the ergodic theorem to copy this behavior to Z . � John Griesmer (U Denver) Sumsets via Spectral Properties June 4 15 / 19
Possible generalizations The proof of the main theorem used equidistribution twice: (i) first to reduce from general measure preserving systems to Kronecker systems, (ii) then to estimate the measure of � A := { n α : n ∈ A } . The perfect squares example shows that we need m ( � A ) > 0 to conclude that A + B is PW-Bohr whenever d ∗ ( B ) > 0. Is this all we need? Question Suppose A ⊂ Z satisfies: A α is dense in Z whenever Z α is dense in Z , meaning A is dense in the Bohr topology of Z . Does this imply d ∗ ( A + B ) = 1 whenever d ∗ ( B ) > 0? John Griesmer (U Denver) Sumsets via Spectral Properties June 4 16 / 19
Possible generalizations Question Suppose A ⊂ Z satisfies: A α is dense in Z whenever Z α is dense in Z , meaning A is dense in the Bohr topology of Z . Does this imply d ∗ ( A + B ) = 1 whenever d ∗ ( B ) > 0? Why this is hard: the hypothesis on A does not imply equidistribution. Example (Katznelson ’73) There is an atomless Borel probability measure σ on [0 , 1] such that � � � � � � > 1 � � A := n : exp(2 π in θ ) d σ ( θ ) � 2 is dense in the Bohr topology on Z . John Griesmer (U Denver) Sumsets via Spectral Properties June 4 17 / 19
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