ISRAEL JOURNAL OF MATHEMATICS xxx (2014), 1–24 DOI: 10.1007/s11856-014-1062-7

D SETS AND ´ ¨ A SARK OZY THEOREM FOR COUNTABLE FIELDS BY

Randall McCutcheon and Alistair Windsor Department of Mathematical Sciences, 373 Dunn Hall The University of Memphis, Memphis, TN 38152-3240, USA e-mail: [email protected] and [email protected]

ABSTRACT

We establish that if F is a countable field of characteristic p, U is a unitary action of F on a Hilbert space H , and P is an essential idempotent ultrafilter on F, then for every polynomial r : F → F with r(0) = 0 one has P-limg Ur(g) f = Pr f weakly, where Pr is the projection onto the closed subspace   Kr = f ∈ H : {Ur(g) f }g∈F is precompact in the strong topology . We then derive combinatorial consequences of this result, including results for sufficiently large finite fields.

1. Introduction S´ ark¨ozy [13] proved that for any set A ⊂ N having positive upper density }| d(A) = lim supN |A∩{1,...,N , (A − A) ∩ {n2 |n ∈ N} = ∅. Utilizing a new correN spondence principle linking density combinatorics with recurrence phenomena in ergodic theory, Furstenberg [9] gave a new proof, deriving it as a consequence of the following theorem. Theorem 1: Let (X, A, μ, T ) be a probability measure preserving system. If r(x) ∈ Z[x] with r(0) = 0 and μ(A) > 0, there exists n ∈ N such that μ(A ∩ T −r(n) A) > 0. Received July 31, 2012 and in revised form January 26, 2013

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We are interested here in versions of Theorem 1 for measure preserving actions of a countable field F. An example of such a result is the following result of P. Larick (cf. [11] or [12]). Theorem 2: Let d ∈ N and let (Tg ) be a measure preserving action of a countable field F on a probability space (X, A, μ). If A ∈ A with μ(A) > 0, then the set R = {g ∈ F : μ(A ∩ Tgd A) > 0} is syndetic. That is, for some finite set F ⊂ F, one has F + R = F. One of the hallmarks of the ergodic method in density combinatorics is that of “largeness of returns”, exhibited here by syndeticity of the set of returns R. Syndeticity, however, fails to capture a crucial property of the sets of returns under consideration, namely the finite intersection property. While it is easy to construct disjoint pairs of syndetic sets (in the integers, even and odd integers provide a natural example), for sets of returns R = {g ∈ F : μ(A ∩ Tgd A) > 0}, S = {g ∈ F : ν(B ∩ Sgd B) > 0} arising from two potentially different measure preserving actions of F, we have that, by Theorem 2, R ∩ S = {g ∈ F : μ × ν((A × B) ∩ (T × S)gd (A × B)) > 0} is again syndetic. This argument extends to any finite number of sets of returns. The method developed by Furstenberg and his coworkers for attributing stronger properties (satisfying the finite intersection property without sacrificing syndeticity) to sets of recurrence times in ergodic theory proceeds by consideration of IP-systems, or what amounts to the same thing, recurrence along arbitrary idempotent ultrafilters. (The etymology of IP is typically traced through “idempotent”.) We review basic facts concerning ultrafilters (see also [10]). An ultrafilter on F is a subfamily P of the power set of F that is closed under finite intersections and supersets, does not contain ∅, and is such that for any A ⊂ F, if A ∈ P then Ac ∈ P. Denote the set of ultrafilters on F by βF. For A ⊂ F put A = {P ∈ βF : A ∈ P}. The sets A form a basis for a compact Hausdorff topology on βF for which the associative operation (P, Q) → P + Q is right continuous, where A ∈ P + Q if and only if {x ∈ F : −x + A ∈ P} ∈ Q. (That is, if Qα → Q then P + Qα → P + Q.) According to a theorem of Ellis

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[6], there are therefore idempotent ultrafilters P. A set R ⊂ F is called IP∗ if it is a member of every idempotent ultrafilter. The family of IP∗ sets has the finite intersection property (see, for example, [10]), and it is easy to show that any IP∗ set is syndetic. One would now like for the following to be true. Conjecture 3: Let d ∈ N and let (Tg ) be a measure preserving action of a countable field F on a probability space (X, A, μ). If A ∈ A with μ(A) > 0 then the set R = {g ∈ F : μ(A ∩ Tgd A) > 0} is IP∗ . We are unable to resolve Conjecture 3 here; efforts to adapt extant proofs seem to run (potentially) afoul of a known class of counterexamples (see [5]) that cause a crucial step in the argument to fail. Fortunately, however, by lowering our sights only slightly we are able to utilize a more recently developed methodology (see [2]) to circumvent the relevant stumbling blocks. Recall that an additive Følner sequence for F is a sequence (Φn ) of finite subsets of F having the property that |Φn (Φn + g)| =0 n→∞ |Φn | lim

for every g ∈ F. Since F is a countable abelian additive group it admits a Følner sequence. We note that each set in a Følner sequence may be independently translated and the resulting sequence is again Følner. Given a Følner sequence (Φn ) for F and A ⊂ F we may define the upper density of A along (Φn ), denoted d(Φn ) (A), by d(Φn ) (A) = lim sup n→∞

|A ∩ Φn | . |Φn |

We define the upper Banach density of A ⊂ F, denoted d∗ (A), by d∗ (A) =

sup (Φn ) Følner

d(Φn ) (A).

An ultrafilter P on an abelian group G, + is an essential idempotent if (1) the upper Banach density of any element of P is positive, i.e., if A ∈ P then d∗ (A) > 0, and (2) P + P = P. A set A ⊆ G is called a D set if there exists an essential idempotent P on G such that A ∈ P. In [1] it is shown that D sets are similar in many ways to so-called central sets. A comparison of various notions of largeness for sets of integers, including D and D ∗ sets, may be found in [4].

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We recall that given a sequence (ag )g∈G ⊂ C we say that P-limg ag = a if for all  > 0, {g ∈ G : |ag − a| < } ∈ P. Given a sequence of functions fg in a w Hilbert space H we say P-limg fg = f weakly, and write P-limg fg = f , if for all h ∈ H, P-limg fg , h = f, h . Idempotence of P implies that if f is any function from G into a compact space, then P-limg P-limh f (g + h) = P-limg f (g).

(1)

We define a set A to be a D∗ set if it has non-trivial intersection with every D set. It is worth noting that in fact the intersection of A with a D set B must again be a D set since if P is the essential idempotent ultrafilter that contains B, then we have either A ∩ B ∈ P or Ac ∩ B ∈ P, but if Ac ∩ B ∈ P, then we have a D set that does not intersect A, so A ∩ B ∈ P. A set A ⊂ F is called syndetic if there is a finite set W such that A+W = F. A set A ⊂ F is called thick if for every finite set W there exists g ∈ F such that g + W ⊂ A. Since a set that is not syndetic has a complement that is thick, a set that has a non-trivial intersection with every thick set is syndetic. By definition, a D∗ set has a non-trivial intersection with every D set. Thus if we show that every thick set is a D set, then we show that every D∗ set is syndetic. Let A ⊂ F be a thick set. Let (Φn ) be a Følner sequence for F. Since A is a thick set, we may translate each Φn to obtain Φn ⊂ A and the new sequence (Φn ) is again a Følner sequence. Let E be the family of subsets E ⊂ F such that d(Φn ) (A \ E) = 0; E is a filter and is therefore contained in an ultrafilter P. We claim that if B ∈ P then d∗ (B) > 0. Suppose that B ∈ P and d∗ (A ∩ B) = 0; then d∗ (A \ B c ) = 0 so d(Φn ) (A \ B c ) = 0 and consequently B c ∈ P. This is a contradiction, so d∗ (A ∩ B) > 0 for all B ∈ P and in particular d∗ (B) > 0, so P is an essential ultrafilter. Since Φn ⊂ A for every g ∈ F we have A \ (A + g) ∩ Φn = Φn \ (A + g) ⊂ Φn \ (Φn + g) ⊂ Φn (Φn + g), and consequently 0 ≤ lim sup n→∞

|A \ (A + g) ∩ Φn | |Φn (Φn + g)| ≤ lim =0 n→∞ |Φn | |Φn |

by the Følner condition. This means A + g ∈ E and hence A + g ∈ P. The operation of ultrafilter addition in βF is right continuous, so the set P + βF is the continuous image of a compact set, hence compact, and it is a semigroup. Thus, by a result of Ellis [6], P+βF contains an idempotent P+Q. If B ∈ P+Q,

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then B − g ∈ P for some g ∈ F and d∗ (B) = d∗ (B − g) > 0, so P + Q is an essential idempotent. Since A + g ∈ P for all g ∈ F we have A ∈ P + Q, so A is a D set. Utilizing weak convergence of certain sequences along essential idempotent ultrafilters, we are able to show that for any d ∈ N, if (Tg ) is a measure preserving action of a countable field F on a probability space (X, A, μ), then for any A ∈ A with μ(A) > 0, the set R = {g ∈ F : μ(A ∩ Tgd A) > 0} is D∗ . Indeed, we will obtain similar results for more general vector space actions and for more general polynomials. Here is our main theorem. Theorem 4: Let F be a countable field of characteristic p, U a unitary action of F on a Hilbert space H , and P an essential idempotent ultrafilter on F. For every polynomial r : F → F with r(0) = 0, one has (2)

P-limg Ur(g) f = Pr f weakly,

where Pr is the projection onto the closed subspace   Kr = f ∈ H : {Ur(g) f }g∈F is precompact in the strong topology . We will postpone the proof of the main theorem until the end of the paper and continue with examining the consequences of the theorem. The following recurrence theorem, for example, can be obtained as a corollary. Corollary 5: Let F be an countable field of characteristic p, (Ty ) a measurepreserving action of F on a probability space (XB, μ), and P an essential idempotent ultrafilter on F. For every polynomial r : F → F with r(0) = 0 and A ∈ B, P-limg μ(A ∩ Tr(g) A) ≥ μ(A)2 . Proof. Given a measure-preserving action we can define a unitary action by Uy f = f ◦ Ty . Applying Theorem 4 to the indicator function of the measurable set A we have P-limg Ur(g) ½A = Pr ½A weakly. Thus P-limg Ur(g) ½A , ½A = Pr ½A , ½A = Pr ½A , Pr ½A = Pr ½A , Pr ½A 1, 1 ≥ Pr ½A , 1 2 = μ(A)2 .

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(Note that the definition of Uy implies that Uy f, 1 = f, 1 , from which it immediately follows that Pr ½A , 1 = ½A , 1 = μ(A).) Theorem 6: Let  > 0, let F be a countable field, and r : F → F a polynomial with r(0) = 0. If E ⊆ F with d∗ (E) > 0, then {g ∈ F : d∗ (E ∩ (E + r(g))) > d∗ (E)2 − } is a D∗ set in F. Proof. Choose a Følner sequence in F , {Φn }, such that lim sup n→∞

|Φn ∩ E| = d∗ (E). |Φn |

Passing to a subsequence if necessary we may suppose that lim

n→∞

|Φn ∩ E| = d∗ (E). |Φn |

By the Furstenberg correspondence principle we have a compact Hausdorff space ΩE endowed with a measure μ and a measure preserving action T of F such that, for some set A ⊂ ΩE with μ(A) = d∗ (E), one has μ(A ∩ Tz−1 A) ≤   d∗ E ∩ (E + z) for all z ∈ F . Let B ⊂ F be an arbitrary D set. By definition there must be an essential idempotent P with B ∈ P. Applying Corollary 5 we immediately obtain P-limg μ(A ∩ Tr(g) A) ≥ μ(A)2 , which implies that {g ∈ F : μ(A ∩ Tr(g) A) ≥ μ(A)2 − } ∈ P. Recasting the above statement in terms of density yields {g ∈ F : d∗ (E ∩ (E + r(g))) ≥ d∗ (E)2 − } ∈ P. Thus, since P is closed under finite intersections, {g ∈ F : d∗ (E ∩ (E + r(g))) ≥ d∗ (E)2 − } ∩ B = ∅. Since B was an arbitrary D set this shows that {g ∈ F : d∗ (E ∩ (E + r(g)) ≥ d∗ (E)2 − } is a D∗ set. An immediate corollary is

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Corollary 7: Let t ∈ N. Suppose that F is a countable field and r : F → F is a polynomial with r(0) = 0. If C ⊂ F is a D set, then C contains a configuration of the form {a, g, a + r(g)}, where g = 0. Proof. By Theorem 6, {g ∈ F : d∗ (C ∩ (C − r(g))) > d∗ (C)2 − } is a D∗ set. Since C is a D set, it follows that R = {g ∈ C : d∗ (C ∩ (C − r(g))) > d∗ (C)2 − } = ∅. Pick g ∈ R and a ∈ (C ∩ (C − r(g))). This result has an immediate finitary consequence. Corollary 8: Let t ∈ N and suppose that F is a countable field, (Φn ) is an increasing Følner sequence in F consisting of additive subgroups, and r : F → F is a polynomial with r(0) = 0. Then there exists N ∈ N such that if ΦN = t i=1 Ci , some Ci contains a configuration {a, g, a + r(g)}, where g = 0.  Proof. Suppose not. For each n ∈ N there is a coloring Φn = ti=1 Ci,n such that no Ci,n contains a configuration of the form {a, g, a + r(g)}, g = 0. Let ⎧ ⎨i if g ∈ C , i,n φn (g) = ⎩0 if g ∈ Φn . Now φn ∈ {0, 1, . . . , t}F , which is compact in the product topology. Let φ ∞ be a limit point of the sequence (φn ). Suppose g ∈ n=1 Φn ; then for all n, Φn ∩(g +Φn ) = ∅ since Φn is an additive subgroup. This contradicts (Φn ) being  F a Følner sequence. Consequently ∞ n=1 Φn = F and we have φ ∈ {1, . . . , t} . Now we may produce a coloring of F by Ci = {g ∈ F : φ(g) = i}. It must be that no Ci contains a configuration of the form {a, g, a + r(g)} with g = 0, since if {a, g, a + r(g)} ∈ Ci for some i, then {a, g, a + r(g)} ∈ Ci,n for sufficiently large n, which contradicts the choice of the Ci,n . This contradicts Corollary 7, t as some Ci is a D set. Thus we must have an N ∈ N such that if ΦN = i=1 Ci , some Ci contains a configuration {a, g, a + r(g)} with g = 0, as required. This finitary result for countable fields allows for a version pertaining to sufficiently large finite fields. Corollary 9: Let t ∈ N and suppose that F is a finite field. Let r(x) ∈ F[x] with r(0) = 0. There is an M such that whenever L is a finite extension of F

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with |L| > M and L = with g = 0.

t i=1

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Ci , some Ci contains a configuration {a, g, a+r(g)}

Proof. Let Φn be an increasing Følner sequence in the countable field of rational functions F(x) consisting of additive subgroups. Let N be as in Corollary 8. Let T be the maximal degree of any polynomial in the numerator or denominator of a member of ΦN . Let M = |F |2T and suppose that L is an extension of F with |L| > M . Since L is a finite extension of a finite field, it is a separable algebraic extension and L = F [x]/j(x) for some irreducible polynomial j(x) ∈ F [x] with deg j(x) > 2T . Now, let y ∈ L have minimal polynomial j(x). Hence for every p(x) ∈ ΦN , consider the evaluation map ey : ΦN → L given by

p(x)  p(y) = ey . q(x) q(y) This map is injective, since if

p (x) 

p (x)  1 2 = ey , ey q1 (x) q2 (x) then p1 (y)q2 (y)−p2 (y)q1 (y) = 0. This requires that p1 (x)q2 (x)−p2 (x)q1 (x) = 0 since the minimal polynomial of y, j(x), has deg j(x) > 2T ≥ deg p1 (x)q2 (x) − p2 (x)q1 (x). Now any t-coloring of L induces a t-coloring of ΦN by pullback under ey . We get a monochromatic configuration {a(x), g(x), a(x) + r(g(x))} ∈ ΦN for the induced coloring, which implies that a(y), g(y), a(y) + r(g(y)) is a monochromatic configuration for the original coloring of L.

2. Proof of Main Theorem 2.1. Polynomial Preliminaries. For r : F → F a polynomial mapping, and j ∈ F, we define Dj r(x) by Dj r(x) = r(x + j) − r(x) − r(j) + r(0). Using this we may define the F-degree of r, denoted degF r, to be the least number n such that Dj1 . . . Djn r = 0 for all j1 , . . . , jn ∈ F. Notice that degF Dj r(x) < degF r(x). We also note that this notion of degree differs from the usual one; we have deg xp = p but degF xp = 1. For all polynomials r(x) we have degF r(x) ≤ deg r(x). We will call a polynomial r(x)

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linear if degF r(x) = 1. Observe that in a field of characteristic p we have l l l (x + y)p = xp + y p , since all the other binomial coefficients are multiples of  p, and thus Dj xp = 0 for all j ∈ F. Indeed a monomial xt is linear if and ∞ only if t = p . More generally, one may show that degF xn = i=1 αi (n) where i n= ∞ i=1 αi (n)p . Given non-empty subsets A, B of an abelian group G, we will say that A generates B if for every b ∈ B there exist a1 , a2 , . . . , aL ∈ A such that g = a1 + · · · + aL . Notice here that L depends on b and need not be bounded. Lemma 10 ([12, Fact 3.2]): Let s1 , . . . , sL be distinct with p  si . {(g s1 , . . . , g sL ) : g ∈ F} generates FL .

Then

This lemma gives no control over the number of terms needed to additively generate. For many of our arguments this will be insufficient and we need a stronger notion. Given non-empty subsets A, B of an abelian group G and a natural number J, we say that A J-generates B if for all b ∈ B there exist a1 , . . . , aL ∈ A with L ≤ J such that b = a1 + · · · + a L . We note that a1 , . . . , aL are not necessarily distinct; L again depends on b but is uniformly bounded by J. We will need the following effective version of [12, Proposition 3.3]. Proposition 11: Consider a polynomial mapping q : F → Fw given by q(x) = (xt1 , xt2 , . . . , xtw ), where 1 < t1 < t2 < · · · < tw and p  ti . There exist R, J ∈ N and a non-zero polynomial Π(x1 , . . . , xR ), both depending on q, such that if k1 , . . . , kR ∈ F satisfy Π(k1 , . . . , kR ) = 0 then {Dki q(g) : g ∈ F, 1 ≤ i ≤ R} J-generates Fw . There is an immediate and important Corollary of this Proposition. Corollary 12: Let r : F → F be a polynomial map given by r(x) = (xt1 , xt2 , . . . , xt ), where 1 ≤ t1 < t2 < · · · < t and p  ti . Then there exists K ∈ N, depending on r, such that {r(g) : g ∈ F} K-generates F .

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Proof of Proposition 11. First we separate the linear exponents from the nonlinear exponents in Dk q(x). We write Dk q(x) = nk (x) + bk (x), where all the non-linear exponents occur in nk (x) and bk (x) =

w

ei

i=1

Ri

ci,j k ti −p

l(i,j)

l(i,j)

xp

,

j=1

where ci,j = 0, l(i, 1) = 0 for 1 ≤ i ≤ w, and e1 = (1, 0, . . . , 0), e2 = (0, 1, 0, . . . , 0), . . . , ew = (0, . . . , 0, 1). w Let u = (u1 , . . . , uw ) ∈ Fw be arbitrary. Let R = i=1 Ri and choose M so that pl(i,j) |M for all i, j. For k1 , . . . , kR ∈ F we consider the system of equations (in variables g1 , . . . , gR ∈ F) ⎧ R ⎨u , j = 1,

l(i,j) l(i,j) i M p ci,j ksti −p gs = 1 ≤ i ≤ w, 1 ≤ j ≤ Ri . ⎩0, j > 1, s=1

Now observe that if g1 , . . . , gR ∈ F solve the above system, then for any h ∈ F we have R

bksM (gs h) =

s=1

R w

ei

s=1 i=1

=

w

ei

Ri

i=1

Ri

ci,j ks(ti −p

l(i,j)

)M

l(i,j)

(gs h)p

j=1 l(i,j)

hp

R

ci,j ks(ti −p

l(i,j)

)M pl(i,j) gs

s=1

j=1

and, since all terms corresponding to j = 1 vanish, therefore (3)

R

bksM (gs h) =

s=1

w

ei hui = hu.

i=1

By our choice of M we have that pl(i,j) |M for all i, j, thus (3) will be true if ⎧ R ⎨u , j = 1

l(i,j) i )M/pl(i,j) ci,j ks(ti −p gs = ⎩0, j > 1 s=1 l(i,j)

where we observe that (ci,j )p = ci,j since ci,j lie in a subfield isomorphic to Fp . This gives us a system of R linear equations in the R variables g1 , . . . , gR (with coefficients that depend on k1 , . . . , kR ). This system will have a solution

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for g1 , . . . , gR if its coefficient matrix (depending on k1 , . . . , kR ) has a non-zero determinant. Letting mi,j = (ti − pl(i,j) )M/pl(i,j) the coefficient matrix is ⎞ ⎛ m m c1,1 k1 1,1 ··· c1,1 kR 1,1 ⎟ ⎜ .. .. ⎟ ⎜ . . ⎟ ⎜ ⎜ m1,R1 m1,R1 ⎟ ⎟ ⎜ c1,R1 k1 · · · c k 1,R1 R ⎟ ⎜ ⎟ ⎜ .. .. A(k1 , . . . , kR ) = ⎜ ⎟. . . ⎟ ⎜ m m ⎜ c k w,1 ··· cw,1 kR w,1 ⎟ ⎟ ⎜ w,1 1 ⎟ ⎜ .. .. ⎟ ⎜ . . ⎠ ⎝ m m cw,Rw k1 w,Rw · · · cw,Rw kR w,Rw We define Π(k1 , . . . , kR ) = det A(k1 , . . . , kR ). Clearly Π is a polynomial in the variables k1 , . . . , kR . Since each constant ci,j occurs in every entry in a row, we  can write det A(k1 , . . . , kR ) = ( i,j ci,j ) det B(k1 , . . . , kR ) where ⎞ ⎛ m m k1 1,1 ··· kR 1,1 ⎜ . .. ⎟ ⎟ ⎜ . . ⎟ ⎜ . ⎜ m1,R1 m1,R ⎟ ⎜ k1 · · · kR 1 ⎟ ⎟ ⎜ ⎜ . .. ⎟ . B(k1 , . . . , kR ) = ⎜ .. . ⎟ ⎟ ⎜ mw,1 ⎟ ⎜ k mw,1 · · · kR ⎟ ⎜ 1 ⎜ . .. ⎟ ⎟ ⎜ . . ⎠ ⎝ . mw,Rw mw,Rw k1 · · · kR This matrix has zero determinant for all k1 , . . . , kR only if mi,j = mi ,j  for some (i, j) = (i , j  ). Now 



ti − pl(i,j) ti − pl( i,j ) = , pl(i,j) pl(i ,j  ) 















ti pl(i ,j ) − pl(i,j) pl(i ,j ) = ti pl(i,j) − pl(i,j) pl(i ,j ) , ti pl(i ,j ) = ti pl(i,j) . However, this is impossible since ti = ti and neither ti nor ti is divisible by p. It remains to deal with the non-linear terms nk . Recall that nk (x) contains all the terms with exponents that are not a power of p. Hence, we may write nk (x) =

w

i=1

ei

Li

j=1

l(i,j)

ai,j (k)xsi,j p

,

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where si,j > 1 and p  si,j . Note that si,j pl(i,j) < ti . Let s1 , . . . , sL be the distinct values of si,j . By Lemma 10 we have (h, hs1 , . . . , hsL ) generates FL+1 , so that there exist h1 , . . . , hT ∈ F such that T

(ht , hst 1 , . . . , hst L ) = (1, 0, . . . , 0) ∈ FL+1 .

t=1

Looking at individual components we see that T

ht = 1,

t=1 T

s

1 ≤ i ≤ w, 1 ≤ j ≤ Li .

ht i,j = 0,

t=1

Thus for any g ∈ F we have T

(4)

nk (ght ) =

t=1

w T

ei

t=1 i=1

=

w

ei

i=1

Li

Li

l(i,j)

ai,j (k)(ght )si,j p

j=1 l(i,j)

ai,j (k)g si,j p

T

s

ht i,j

pl(i,j)

= 0.

t=1

j=1

Now using (3) and (4) we have R T

DksM q(gs ht ) =

t=1 s=1

R T

bksM (gs ht ) +

t=1 s=1

=

T

T R

nksM (gs ht )

s=1 t=1

ht u + 0 = u.

t=1

Since u ∈ F was arbitrary this completes the proof. We see that the uniform constant J = T R. w

Proof of Corollary 12. Define a polynomial q : F → Fw by ⎧ ⎨r(x) = (xt1 , . . . , xt ) if t > 1, 1 q(x) = t t ⎩(x 2 , . . . , x  ) if t1 = 1.

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If t1 = 1 we have w =  − 1, otherwise we have w = . By Proposition 11 there exist R, J ∈ N and a non-zero polynomial Π(x1 , . . . , xR ), all depending on q, such that if k1 , . . . , kR ∈ F satisfy Π(k1 , . . . , kR ) = 0, then {Dki q(g) : g ∈ F, 1 ≤ i ≤ R} J-generates Fw . Since Π is non-zero, there exist k1 , . . . , kR ∈ F such that Π(k1 , . . . , kR ) = 0 and hence {Dki q(g) : g ∈ F, 1 ≤ i ≤ R} J-generates Fw . Let y ∈ Fw be arbitrary. From the definition of J-generating we have that there exist L ∈ N with L ≤ J, 1 ≤ i1 , . . . , iL ≤ R, and g1 , . . . , gL ∈ F such that y=

L

Dkim q(gm ).

m=1

Recalling the definition of Dk q(g) we have y=

L

Dkim q(gm )

m=1

=

L

(q(gm + kim ) − q(gm ) − q(kim ))

m=1

=

L

(q(gm + kim ) + (p − 1)q(gm ) + (p − 1)q(kim )).

m=1

Thus we see that K = (2p − 1)J terms from {q(g) : g ∈ F} suffice to write any y ∈ Fw . If t1 > 1, then q = r and this completes the proof. In the case that t1 = 1, we observe that Dk x = 0 for all k ∈ F and hence   Dk r(x) = 0, Dk q(x) . Let y ∈ F be arbitrary. From the previous argument we have that {Dki q(g) : g ∈ F, 1 ≤ i ≤ R} J-generates Fw , so there exist L ∈ N, with L ≤ J, 1 ≤ i1 , . . . , iL ≤ R, and g1 , . . . , gL ∈ F such that L

(q(gm + kim ) + (p − 1)q(gm ) + (p − 1)q(kim )) = (y2 − y1t2 , . . . , y − y1t ).

m=1

Passing to r(x) and F we have L

(r(gm + kim ) + (p − 1)r(gm ) + (p − 1)r(kim )) = (0, y2 − y1t2 , . . . , y − y1t )

m=1

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and hence r(y1 ) +

L

(r(gm + kim ) + (p − 1)r(gm ) + (p − 1)r(kim )) = y.

m=1

Thus we see that K = (2p − 1)J + 1 terms from {r(g) : g ∈ F} suffice to write any y ∈ F . This completes the proof in the case t1 = 1. The following Lemma makes clear the role of J-generating sets in our argument. Lemma 13: Suppose that A1 , . . . AR , B ⊂ F and that A1 ∪· · ·∪AR J-generates B. Let f ∈ H . If for each 1 ≤ j ≤ R, {Uy f : y ∈ Aj } is precompact in the strong topology, then {Uy f : y ∈ B} is precompact in the strong topology. Proof of Lemma 13. Let  > 0 be arbitrary. Choose {ai,j ∈ Aj : 1 ≤ i ≤ Kj } such that {Uai,j f : 1 ≤ i ≤ Kj } is an -net for {Uy f : y ∈ Aj }. Let b ∈ B be arbitrary. Since A1 ∪ · · · ∪ AR J-generates B, there exist 1 ≤ j1 , . . . jm ≤ R and a1 , . . . , aM ∈ F with am ∈ Ajm such that b = a1 + · · · + a M and M ≤ J. Now for each 1 ≤ m ≤ M choose aim ,jm ∈ Ajm such that Uaim ,jm f − Uam f  < . For m = 1 we have Uai1 ,j1 f − Ua1 f  ≤ . Suppose that we have (5)

Uai1 ,j1 +···+aim ,jm f − Ua1 +···+am f  ≤ m;

then, since U is a unitary action, we may apply Uam+1 to the difference inside the norm in (5) to obtain (6)

Uai1 ,j1 +···+aim ,jm Uam+1 f − Ua1 +···+am +am+1 f  ≤ m.

Similarly, we can apply Uai1 ,j1 +···+aim ,jm to the difference inside the norm in Uaim+1 ,jm+1 f − Uam+1 f  < 

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to obtain (7)

Uai1 ,j1 +···+aim ,jm +aim+1 ,jm+1 f − Uai1 ,j1 +···+aim ,jm Uam+1 f  ≤ .

Combining (6) and (7) yields the desired Uai1 ,j1 +···+aim+1 ,jm+1 f − Ua1 +···+am+1 f  ≤ (m + 1). Therefore, by induction we have Uai1 ,j1 +···+aiM ,jM f − Ua1 +···+aM f  ≤ M  ≤ J or, equivalently, Uai1 ,j1 +···+aiM ,jM f − Ub f  ≤ M  ≤ J. Since b ∈ B was arbitrary, and J is a uniform bound, we therefore have a finite J-net for {Uy f : y ∈ B}. Thus {Uy f : y ∈ B} is precompact. 2.2. Proof of Main Results. We will make use of the following standard lemma: Lemma 14 (Van der Corput): Suppose that P is an idempotent ultrafilter on F and (fg )g∈F is a bounded sequence in a Hilbert space H . If P-limh P-limg fg+h , fg = 0, then P-limg fg = 0. Proof. We observe that as a consequence of (1) we have P-limg fg = P-limg1 . . . P-limgn

n 1 fg +···+gn . n m=1 m

Applying the norm squared and using lower semicontinuity we obtain  n 2  1  f  P-limg fg 2 ≤ P-limg1 . . . P-limgn 2  gm +···+gn   n m=1  n  n

1 = P-limg1 . . . P-limgn 2 fg +···+gn , fgm +···+gn n m=1 m m=1 = P-limg1 . . . P-limgn

n n 1

fgl +···+gn , fgm +···+gn n2 m=1 l=1

n n 1

P-limg1 . . . P-limgn fgl +···+gn , fgm +···+gn . ≤ 2 n m=1 l=1

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Collapsing limits using (1) and separating the cases when l = m we obtain  P-limg fg 2 ≤

n

1 P-limg fg 2 + 2 Re P-limh P-limg fg , fg+h 2 n m=1 l
1 P-limg fg 2 . n Since n was arbitrary this gives P-limg fg = 0. =

Proposition 15: Suppose that the hypotheses of Theorem 4 hold. If for f ∈ H we have {Uy f : y ∈ F } is precompact in the strong topology, then P-limg Ur(g) f = f. Proof of Proposition 15. As a consequence of the spectral theorem the Kronecker subspace of the unitary action of F  is spanned by eigenfunctions, namely by functions h for which there is a character χ : F → S 1 where S 1 = {z ∈ C : |z| = 1} such that Uy h = χ(y)h for all y ∈ F . By linearity it suffices to show that P-limg Ur(g) h = h for all eigenfunctions h. Since P-limg Ur(g) h = P-limg χ(r(g))h, we see that this reduces to the statement P-limg χ(r(g)) = 1 for all characters χ. We use induction on degF r(x). Suppose that the result holds for all polynomials with degF r(x) < n. Since S 1 is compact we have that P-limg w(r(g)) exists. Let z = P-limg w(r(g)). Let V ⊂ S 1 be an open neighborhood in S 1 with 1 ∈ V and V = V . Now zV is a neighborhood of z in S 1 and by definition A = {g ∈ F : w(r(g)) ∈ zV } ∈ P. Since P is idempotent, A = {g ∈ A : A − g ∈ P} ∈ P. Fix g ∈ A and observe that Ag = {h ∈ F : w(Dg r(h)) ∈ V } ∈ P by our induction hypothesis. Since g ∈ A , we have A−g ∈ P so Ag ∩(A−g)∩A ∈ P and consequently we can let h ∈ Ag ∩ (A − g) ∩ A = ∅. Thus, by definition of

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A − g we have that g + h ∈ A. Since r(0) = 0 we have Dg r(h) + r(g) + r(h) = r(g + h) and consequently w(r(g + h)) = w(Dg r(h))w(r(h))w(r(g)). Now g ∈ A ⇒ w(r(g)) ∈ zV, h ∈ A ⇒ w(r(h)) ∈ zV, h ∈ Ag ⇒ w(Dg r(h)) ∈ V, and thus w(r(g + h)) ∈ z 2 V 3 . However, g + h ∈ A ⇒ w(r(g + h)) ∈ zV. Thus 1 = w(r(g + h))w(r(g + h)) ∈ zV z 2 V 3 = zV 4 , and since V was an arbitrary symmetric neighborhood of 1 we must have z = 1. We have the following crucial Corollary that shows that the Main Theorem holds for all f in Kr . Corollary 16: Suppose that the hypotheses of Theorem 4 hold. Let r : F → F be a polynomial map given by r(x) = (xt1 , xt2 , . . . , xt ), where 1 ≤ t1 < t2 < · · · < t and p  ti . If f ∈ Kr , i.e., {Ur(g) f : g ∈ F} is precompact in the strong topology, then P-limg Ur(g) f = f. Proof of Corollary 16. Suppose that {Ur(g) f : g ∈ F} is precompact. From Corollary 12 we have {r(g) : g ∈ F} K-generates F . Now from Lemma 13 we have {Uy f : y ∈ F } is precompact. Hence by Proposition 15 we have P-limg Ur(g) f = f . Lemma 17: Suppose that the hypotheses of Theorem 4 hold. Let r : F → F be a polynomial map given by r(x) = (x, xt2 , . . . , xt ),

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  where 1 < t2 < · · · < t and p  ti . If Uy f : y ∈ {0} × F−1 is precompact in the strong topology then P-limg Ur(g) f = Pr f . Proof. Let   K2 = h ∈ H : {Uy f : y ∈ {0} × F−1 } is precompact ; K2 is a Uy invariant subspace. Since the first coordinate of r is linear, we see that all derivatives of r have 0 in the first coordinate and hence {Dk r(g) : g ∈ F} ⊂ {0} × F−1 and as a consequence K2 ⊂ KDj r for all j ∈ F. w For h ∈ K2 we define an operator h by Sh = P-limg Ur(g) h. We will show that for h ∈ K2 we have Sh = Pr h. First we show that S is indeed a projection onto a subspace of K2 . Indeed, S 2 h = P-limg Ur(g) Sh = P-limg P-limg Ur(g) Ur(g ) h, and since h ∈ KDg r we have P-limg UDg r(g ) h = h. Since there is no loss of norm, the weak convergence implies strong convergence and we may write S 2 h = P-limg P-limg Ur(g) Ur(g ) UDg r(g ) h = P-limg P-limg Ur(g+g ) h = P-limg Ur(g) h = Sh and S ≤ 1, so S is an orthogonal projection. If f ∈ Kr then, by Corollary 16, P-limg Ur(g) f = f or, equivalently, Sf = f . This shows range Pr ⊂ range S. Hence we will be done once we show that range S ⊂ range Pr . Now h ∈ range S if h ∈ K2 and Sh = P-limg Ur(g) h = h. Since there is no loss of norm, weak convergence implies strong convergence. Thus, for  > 0, by definition of P-lim, we have E = {g ∈ F : Ur(g) h − h < } ∈ P. Since P is an essential ultrafilter we have d∗ (E) > 0. As Følner observes [8, 7] d∗ (E) > 0 implies that E − E is syndetic, i.e., there exists a finite set W ⊂ F such that (E − E) + W = F. To see this, suppose that W = {w1 , . . . , wn } ⊂ F is a finite set in F with the property that (E − wi ) ∩ (E − wj ) = ∅ for all i, j ∈ W with i = j and which is maximal in the sense that there does not

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exist a wn+1 ∈ F such that {w1 , . . . , wn+1 } has the same property. This is   1 possible since d∗ (E − w1 ) ∪ · · · ∪ (E − wn ) = n · d∗ (E) ≤ 1 so that n ≤ d∗ (E) . By maximality of W , for g ∈ F arbitrary there exists w(g) ∈ W such that (E − g) ∩ (E − w(g)) = ∅ so that g ∈ (E − E) + w(g). In particular, F = (E − E) + W . Let W = {w1 , . . . , wn } be such a finite set in F. Since h ∈ K2 we have {U(0,g2 ,...,g ) h : g2 , . . . , g ∈ F} is precompact, hence for each 1 ≤ i ≤ n the isometric image {U(wi ,g2 ,...,g ) h : g2 , . . . , g ∈ F} is precompact, and thus the finite union {U(w,g2 ,...,g ) h : w ∈ W, g2 , . . . , g ∈ F} is precompact. Let N be an -net for {U(w,g1 ,...,g ) h : w ∈ W, g2 , . . . , g ∈ F}. Given g ∈ F we may write g = x − y + w where x, y ∈ E and w ∈ W . Now r(g) = (x − y + w, g t2 , . . . , g t ) = r(x) − r(y) + (w, g2 , . . . , g ), where gi = g ti − xti + y ti . Choose φ ∈ N such that U(w,g2 ,...,e ) h − φ < . Now Ur(g) h − φ ≤Ur(g) h − U−r(y)+(w,g2 ,...,g ) h + U−r(y)+(w,g2 ,...,g ) h − U(w,g2 ,...,g ) h + U(w,g2 ,...,g ) h − φ ≤Ur(x)h − h + Ur(y)h − h + U(w,g2 ,...,g ) h − φ ≤3. Since  > 0 was arbitrary we see that this means that {Ur(g) h : g ∈ F} is precompact and h ∈ range Pr . Proof of Main Theorem. We use induction on degF r(x). If degF r(x) = 1, then Ur(g) is an action and Theorem 4 is [3, Corollary 4.6]. Suppose that the conclusion of the theorem holds for all r(x) with degF r(x) < d. We need to show that the conclusion holds for all r(x) with degF r(x) = d. We begin by considering a monomial r : F → F of the form r(x) = (xt1 , xt2 , . . . , xt ),

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where 1 ≤ t1 < t2 < · · · < t and p  ti , and degF r = d. We shall say that an element f ∈ H is r-good if (2) holds for f . Similarly, we shall say that a closed linear subspace D ⊂ H is r-good if every f ∈ D is r-good. Clearly the set of f ∈ H which are r-good is closed. Thus given any increasing chain in the space of r-good subspaces we have an upper bound. Thus by Zorn’s lemma we must have a maximal r-good subspace. We shall denote this maximal r-good subspace by Hm . Let Q denote the orthogonal projection on Hm . To complete the proof it will suffice to find 0 = f ∈ Hm⊥ such that f is r-good. We claim that for all y ∈ F the action Uy commutes with the projection Pr , i.e., (8)

Uy Pr = Pr Uy .

Observe that Kr is U invariant. Since U is a unitary action we must have Kr⊥ is U -invariant. For any f ∈ H we may uniquely decompose f = fr + ft⊥ , where fr ∈ Kr and fr⊥ ∈ Kr⊥ . Observe that by invariance Uy f = Uy fr + Uy fr⊥ . Thus for all f ∈ H we have Pr Uy f = Pr (Uy fr + Uy fr⊥ ) = Uy fr = Uy Pr f. An immediate consequence of (8) is that for all j1 , j2 ∈ F PDj1 r PDj2 r = PDj2 r PDj1 r . We next claim that for all y ∈ F the maximal space Hm is Uy invariant. Indeed for any f ∈ Hm and h ∈ H we have P-limg Ur(g) Uy f, h = P-limg Ur(g) f, Uy−1 h = Pr f, Uy−1 h = Uy Pr f, h = Pr Uy f, h . Now that we know that Hm is U invariant, the same argument as above shows that Uy commutes with the orthogonal projection on Hm , i.e., for all y ∈ F (9)

Uy Q = Q Uy .

By our induction hypothesis we have that for all j ∈ F H is Dj r-good, i.e., for all f ∈ H w

P-limg UDj r(g) f = PDj r f.

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We claim that if f ∈ Hm⊥ then PDj r f ∈ Hm⊥ or, equivalently, if Qf = 0 then QPDj r f = 0. Indeed, if for all h ∈ H f, Qh = 0, then PDj r f, Qh = P-limg UDj r(g) f, Qh −1 Qh = P-limg f, UD j r(g) −1 h = P-limg f, QUD j r(g)

by (9)

= 0. Thus PDj r f ∈ Hm⊥ . Now let 0 = f0 ∈ Hm⊥ and let Π(x1 , . . . xR ) be the polynomial given by Proposition 11. Let H0 = {j ∈ F : Π(j, x2 , . . . , xR ) = 0}; H0 is cofinite. If Pj f0 = 0 for all j ∈ H0 , then let H = H0 and f = f0 and stop. Otherwise, pick j1 ∈ H0 such that f1 = Pj1 f0 = 0. Observe that f1 ∈ Hm⊥ . Having chosen j1 , . . . , js ∈ F with s < R such that fs = PDjs r . . . PDj1 r f0 = 0 we let Hs = {j ∈ F : Π(j1 , . . . , js , j, xs+2 , . . . , xR ) = 0}. If PDj r fs = 0 for all j ∈ Hs then we stop. Otherwise, we pick js+1 ∈ Hs+1 such that fs+1 = PDjs+1 r fs = 0 and continue. There are two cases to consider. Case 1: The process terminates for some s < R. We let f = fs . Now 0 = f ∈ Hm⊥ and for all j ∈ Hs we have PDj r f = 0. Let fg = Ur(g) f . For all j ∈ Hs , a cofinite set, we have P-limg fg , fg+j = P-limg Ur(g) f, Ur(g+j) f = P-limg Ur(g) f, Ur(g) Ur(j) UDj r(g) f = P-limg f, Ur(j) UDj r(g) f −1 f, UDj r(g) f = P-limg Ur(j) −1 f, PDj r f = P-limg Ur(j)

using our inductive hypothesis. However, by construction PDj r f = 0, so we have P-limg fg , fg+j = 0

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for all but finitely many j ∈ F. It follows from the van der Corput Lemma, Lemma 14, that P-limg Ur(g) f = 0. We now claim that Pr f = 0. Let f = fr + fr⊥ ∈ Kr ⊕ Kr⊥ . Recall that the splitting Kr ⊕ Kr⊥ is Uy -invariant and hence P-limg Ur(g) fr ∈ Kr and P-limg Ur(g) fr⊥ ∈ Kr⊥ . Since P-limg Ur(g) f = P-limg Ur(g) fr +P-limg Ur(g) fr⊥ = 0 we must have P-limg Ur(g) fr = 0. However, we know from Corollary 16 that P-limg Ur(g) fr = fr and thus we have Pr f = fr = 0, as required. Thus we have P-limg Ur(g) f = Pr f and f ∈ Hm , which is a contradiction. Case 2: The process does not terminate for any s < R. In this case we let f = fR = PDjR r . . . PDj1 r f0 = 0 and have Π(j1 , . . . , jR ) = 0. Since the projections PDji r commute, we have f ∈ KDji r for 1 ≤ i ≤ R and therefore by definition {UDji r(g) f : g ∈ F} is precompact for 1 ≤ i ≤ R. Case 2.1 [t1 > 1]. We have, by Corollary 12, that {Dji r(g) : g ∈ F, 1 ≤ i ≤ R} J-generates F . Applying Lemma 13 we have that {Uy f : y ∈ F } is precompact. This implies that f ∈ Kr and hence f = Pr f . Moreover, by Lemma 15, we have that P-limg Ur(g) f = f . Thus P-limg Ur(g) f = Pr f . This implies that f ∈ Hm and hence we have a contradiction. Case 2.2 [t1 = 1]. We have, by Corollary 12, that {Dji r(g) : g ∈ F, 1 ≤ i ≤ r} J-generates {0} × F−1 . Applying Lemma 13 we have that {Uy f : y ∈ {0} × F−1 } is precompact. By Lemma 17 we have P-limg Ur(g) f = Pr f . This implies that f ∈ Hm and hence we have a contradiction. In all the cases we arrive at a contradiction. Therefore, there cannot exist an 0 = f ∈ Hm⊥ and we must have Hm = H so that Theorem 4 holds for all f ∈ H when r : F → F is a monomial of the form r(x) = (xt1 , xt2 , . . . , xt ), where 1 ≤ t1 < t2 < · · · < t and p  ti and degF r = d . The extension from such monomials to general polynomials follows the same process as in [12]. For completeness we give that process here. Suppose that r : F → F is a general polynomial mapping r(x) =



i=1

ei

Ri

i=1

ci,j xti,j

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with degF r = d. Let ti,j = si,j pl(i,j) where p  si,j . Let s1 < · · · < sL be the distinct values of si,j . Define    Ri

l(i,j) . ei ci,j g p Vg(k) = U i=1 l(i,j)

Since g p

j=1 si,j =sk

is linear this is a unitary action of F. We define V(g1 ,...,gL ) =

L 

Vg(k) , k

k=1 (1)

(L)

commuting V(g1 ,...,gL ) is a unitary action of F . since every V , . . . , V Let rˆ(g) = (g s1 , . . . , g sL ) and observe that Vrˆ(g) = V

(gs1 ,...,gsL )

=

L 

U

 

 

ei

i=1

=U

 

ei

i=1

Ri

Ri

ci,j g

si,j pl(i,j)

j=1 si,j =sk

Ri L

k=1

ei

(k)

Vgsk

k=1

i=1

k=1

=U

=

L 

l(i,j)





ci,j g si,j p

j=1 si,j =sk l(i,j)



ci,j g si,j p

j=1

= Ur(g) . Thus the polynomial result for U can be obtained by applying our monomial result to the action V with the monomial rˆ(g) since degF rˆ ≤ d. Thus we conclude that Theorem 4 holds for all polynomials r with degF r(x) ≤ d. By induction it therefore holds for all polynomials r.

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[3] V. Bergelson, Minimal idempotents and ergodic Ramsey theory, in Topics in Dynamics and Ergodic Theory, London Mathematical Society Lecture Note Series, Vol. 310, Cambridge University Press, Cambridge, 2003, pp. 8–39. [4] V. Bergelson and T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure preserving systems, Colloquium Mathematicum 110 (2008), 117–150. [5] V. Bergelson, H. Furstenberg and R. McCutcheon, IP-sets and polynomial recurrence, Ergodic Theory and Dynamical Systems 16 (1996), 963–974. [6] R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969. [7] E. Følner, Generalization of a theorem of Bogolio` uboff to topological abelian groups. With an appendix on Banach mean values in non-abelian groups, Mathematica Scandinavica 2 (1954), 5–18. [8] E. Følner, Note on a generalization of a theorem of Bogolio` uboff, Mathematica Scandinavica 2 (1954), 224–226. [9] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemer´ edi on arithmetic progressions, Journal d’Analyse Math´ ematique 31 (1977), 204–256. ˇ [10] N. Hindman and D. Strauss, Algebra in the Stone–Cech Compactification. Theory and Applications, Second revised and extended edition, de Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. [11] P. G. Larick, Results in polynomial recurrence for actions of fields, Thesis (Ph.D.)–The Ohio State University, ProQuest LLC, Ann Arbor, MI, 1998. [12] R. McCutcheon, A S´ ark¨ ozy theorem for finite fields, Combinatorics, Probability and Computing 12 (2003), 643–651; Special issue on Ramsey theory. [13] A. S´ ark¨ ozy, On difference sets of sequences of integers. III, Acta Mathematica Academiae Scientiarum Hungaricae 31 (1978), 355–386.