Polynomial configurations in sets of positive upper density over local fields

0.5 MB·

by  Mohammad Bardestani

POLYNOMIAL CONFIGURATIONS IN SETS OF POSITIVE UPPER DENSITY
OVER LOCAL FIELDS
MOHAMMAD BARDESTANI AND KEIVAN MALLAHI-KARAI
Abstract. Let K be the field of real or p-adic numbers, and F(x) = (f (x),...,f (x)) be such
1 m
that 1,f ,...,f are linearly independent polynomials with coefficients in K. Employing ideas of
1 m
Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory integrals
withpolynomialphasewewillshowthattheindependenceratiooftheCayleygraphofKmwithrespect
to the portion of the graph of F defined by a ≤ log|s| ≤ T is at most O(1/(T −a)). From here, we
conclude that if I ⊆ Km has positive upper density, then the difference set I −I contains vectors of
7
1 the form F(s) for an upbounded set of values s∈K. We deduce that the Borel chromatic number of
0 the Cayley graph of Km with respect to the set {±F(s) : s ∈ K} is infinite, while the clique number
2 of this Cayley graph is finite. Moreover, the infinitness of the Borel chromatic number does not hold
necessarily if f ,...,f are merely real-analytic functions.
n 1 m
a
J

0
Contents
3

1. Introduction 1
]
O
2. Main results 3
C 3. Preliminaries 6
. 4. A spectral bound for Independence ratio 7
h
t 5. Oscillatory integrals with real polynomial phase 11
a
m 5.1. Low frequency case 12
5.2. High frequency case 13
[
6. Oscillatory integrals with p-adic polynomial phase 15

2
7. Affine B´ezout theorem and Proof of Theorem 2.11 17
v
4 8. Some remarks on Cayley graphs of curves 18
2 9. Algebraic aspects of Cayley graphs 19
0
Acknowledgement 20
6
References 20
0
.
1
0
7
1 1. Introduction
:
v Let A be a non-compact abelian topological group, equipped with an invariant distance d. Fix a
i
X symmetric subset S ⊆ A that doesn’t contain the identity element of A. The following two related
r problems (for A = Rn or A = Zn, and various choices of S and Ω) have been the subject of many
a
works:
1) (Finding configurations) Suppose Ω ⊆ A has positive upper density, that is, Ω contains a
definite proportion of an increasing sequence of d-balls. When does the equation x−y = s
have a solution with x,y ∈ Ω and s ∈ S?
2) (Efficient coloring)LetCay(A,S)denotetheCayleygraphofAwithrespecttoS. Whenisthe
chromatic (or Borel chromatic) number of Cay(A,S) finite? When yes, establish non-trivial
lower and upper bounds.
Let us review some of the existing literature on these themes. Furstenberg, Katznelson and Weiss [17]
used tools from ergodic theory to prove that if the upper density of Ω ⊆ R2 is strictly positive then
for all sufficiently large real numbers (cid:96) one can find points x,y ∈ Ω with euclidean distance (cid:96) from
Keywords: Chromatic number; Fourier transform; Local fields, Oscillatory integrals.
2010 Mathematics Subject Classification: Primary 05C90; Secondary 47A10.
1
2 M.BARDESTANIANDK.MALLAHI-KARAI
each other i.e., x−y ∈ (cid:96)S1 for any large enough (cid:96). This subject, known as Geometric Ramsey theory,
has been the source of many investigations; we refer the reader to Ziegler [32] for a development
of this theory. Falconer and Marstrand [14] obtained another proof of Furstenberg, Katznelson and
Weiss’ theorem by means of geometric measure theory. Subsequently, Bourgain [8] applied ideas from
harmonic analysis to generalize the theorem to Rm. A probabilistic proof is also given recently by
Quas[22]. Moreover,aquantitativeversionoftheseresultshasbeenobtainedbyBukh[9]andOliveira
and Vallentin [11]. The euclidean norm is obtained by the unit ball and one might wonder what would
happen if the unit ball is replaced with a convex body K. Notice that K produces a new norm defined
by
(cid:107)x(cid:107) := sup{t ≥ 0 : tx ∈ K}.
K
This problem has been considered by Kolountzakis [19] who proved a result analogues to the theorem
of Furstenberg, Katznelson, Weiss, Falconer, Marstrand and Bourgain when K is essentially not a
polytope. We refer the reader to the original paper for more details.
Similar problems, involving polynomials, have been studied for A = Z. A theorem of Furstenberg
and S´ark¨ozy states that if p(x) ∈ Z[x] and p(0) = 0, then in any set of positive upper density one can
find distinct elements x,y in the given set such that the equation
x−y = p(n),
has a solution for some n ∈ Z. Bergelson and Leibman [4] (extending on the ideas of Furstenberg)
established a far–reaching qualitative generalization of Furstenberg-S´ark¨ozy’s theorem, often refered
to as the Polynomial Szemer´edi Theorem. This theorem has been further extended to more general
polynomialsin[5]. SubsequentlyLyallandMagyar[21]obtainedaquantitativeversionofFurstenberg-
S´ark¨ozy’stheorem. Moreprecisely, forA ⊆ [1,N]∩Zandafamilyoflinearlyindependentpolynomials
p (x),...,p (x) in Z[x] with p (0) = 0, Lyall and Magyar proved that if
1 m i
(1) {p (d),...,p (d)} (cid:54)⊆ A−A,
1 m
for any 0 (cid:54)= d ∈ Z, then
|A| (cid:18)(loglogN)2(cid:19)1/m(k−1)
(2) ≤ C , k := max degp ,
i
N logN 1≤i≤m
for some absolute constant C = C(p ,...,p ). This result is a direct consequence of higher dimen-
1 m
sional analogue of Furstenberg-S´ark¨ozy’s theorem [21, Theorem 3]. For a qualitative version of this
result, obtained by ergodic theoretical method, see [6, Corollary 4.2.1]
Let us now turn to the second theme. The most famous question of this type asks about the
chromatic number of the unit distance graph, the Cayley graph of Rm with respect to the unit sphere.
In dimension two, this is known as the Hadwiger-Nelson problem and has a long history. It has been
known for a long time that
4 ≤ χ(Cay(R2,S1)) ≤ 7.
The upper bound is proved via a hexagonal partitioning of the plane and the upper bound can be es-
tablishedbyconsideringaspecific7-vertexsubgraphknownasMoserspindle. Answeringaffirmatively
a question of Erd˝os, Frankl and Wilson [16] proved that for m ≥ 3,
cm ≤ χ(Cay(Rm,Sm−1)).
1
for c = 1.207 + o(1). This was subsequently improved to c = 1.239 + o(1) by Raigorodskii [23].
1 1
Larman and Rogers [20] proved the upper bound χ(Cay(Rm,Sm−1)) ≤ cm, for c = 3+o(1). For more
2 2
regarding this problem, we refer the reader to [27, 29].
Let A and S be as above, and G = Cay(A,S) denote the Cayley graph of A with respect to S.
The Borel chromatic number of G, denoted by χ (G), is the least cardinal c such that the set
Bor
A can be partitioned into c Borel subsets none of which contains two adjacent vertices of G. One of
the advantages of working with the Borel chromatic number is that it allows for analytic tools to be
deployed. The study of the measurable chromatic number was initiated by Falconer [13] who proved
that
χ (Cay(Rm,Sm−1)) ≥ m+3.
Bor
Falconer’s method is based on geometric measure theory. Recently, by bringing new ideas from
Fourier analysis and linear programming method, Bachoc, Nebe, Oliveira and Vallentin [2], among
BOREL CHROMATIC NUMBER OF CAYLEY GRAPHS 3
other things, extended significantly the lower bound of χ (Cay(Rm,Sm−1)) when 10 ≤ m ≤ 24.
Bor
These bounds have been further extended by Oliveira and Vallentin [11] to the range 3 ≤ m ≤ 24.
Another variation of the unit distance graph is the following graph considered by Sz´ekely [29]. Let
H be a set of positive real numbers. Sz´ekely introduced the graph G with the vertex set Rm in which
H
two points are joined if their euclidean distance is in H and conjectured that if H is not bounded from
above then the Borel chromatic number is infinite. Note that the theorem of Furstenberg, Katznelson
and Weiss [17] mentioned above answers this question positively.
One of the technique for lower-bounding the chromatic number of finite graphs involves spectral
analysis of the graph. The classical result in this direction is due to Hoffman who gave a lower bound
for the chromatic number of a finite graph G in terms of the smallest and largest eigenvalues of the
adjacency matrix of G. Steinhardt [28], by providing a version of Hoffman’s spectral bound for infinite
topologicalgraphs,wasabletogiveaspectralproofofinfinitenessoftheBorelchromaticnumberofodd
distance graph which is defined by Cay(R2,∪ (2k+1)S1). We remark that the odd distance graph,
k≥0
is a special case of Sz´ekely’s construction when H is the set of all positive odd integers. Steinhardt’s
method has been generalized systematically by Bachoc, DeCorte, Oliveira and Vallentin [1]. Notice
that for any topological graph G the obvious bound χ(G) ≤ χ (G) holds. To indicate the power of
Bor
analytic tools, let us mention that it is an open problem whether the ordinary chromatic number of
the odd distance graph is infinite.
All the graphs described so far are Cayley graphs with respect to a family of spheres of various
radii. Hence these graphs are defined by algebraic relations. In our previous work [3], by applying
the spectral method developed by Bachoc, DeCorte, Oliveira and Vallentin [1], we showed that the
Borel chromatic number of the Cay(K2,Σ) = ℵ , where K is a local field of characteristic zero and
0
Σ = {(x,y) ∈ K2 : xy = 1}. As an application we proved that if SL (K), is partitioned into
2
finitely many Borel sets then in at least one of these sets one can find two matrices A,B such that
det(A+B) = 0. Motivated by this, it is natural to ask for a classification of real and p-adic algebraic
sets in Rm or Qm in which the Borel chromatic number of their associated Cayley graphs is infinite.
p
Question 1. Let K be either R or Q . Classify those Zariski closed sets V ⊆ Am such that
p K
(3) χ (Cay(Km,±V)),
Bor
is infinite.
Our interest in this paper is to investigated Question 1. We will develop techniques for estab-
lishing “almost non-negativity” bounds for certain oscillatory integrals with polynomial phase and
subsequently apply these bounds to combinatorial problems of the types indicated above. As a result,
pertaining to finding configuration and effective coloring, we will obtain non-trivial upper bounds for
theindependencedensityofcertainCayleygraphsandlowerboundsfortheirBorelchromaticnumber.
Interestingly enough, these results are not generally true even for real analytics curves.
2. Main results
Notation. Through this paper we let K denote R or the field of p-adic numbers Q . The vector
p
spaces Rm and Qm are respectively equipped with the euclidean and the supremum norm, both of
p
which will be denoted by (cid:107)·(cid:107). The ball of radius r centered at zero with respect to the metric induced
by these norms is denoted by B . As a locally compact topological group Km has a Haar measure.
r
The measure of a Borel set A ⊆ Km will be denoted by |A|. When K = Q , the measure is normalized
p
so that Zm, the ring of p-adic integers, has measure one. For a set I ⊆ Km, the difference set I −I
p
consists of all differences i −i with i ,i ∈ I.
1 2 1 2
Before stating the results, let us review some terminology. Let S ⊆ Km be an arbitrary Borel set.
The Cayley graph of Km with respect to the symmetrized set ±S, denoted by
G := Cay(Km,±S),
is an infinite graph with vertex set Km and the edge set {{x,x±s} : x ∈ Km,s ∈ S}. Note that the
vertex set of G is also furnished with a Borel structure induced by the topology of K.
4 M.BARDESTANIANDK.MALLAHI-KARAI
Anindependent setofG isaBorel subsetofKm whichdoesnotcontainapairofadjacentvertices.
The upper density of any Borel set I ⊆ Km is defined by
|I ∩B |
(4) d¯(I) := limsup r .
|B |
r→∞ r
The independence ratio of G is defined by
(5) α¯(G) := sup(cid:8)d¯(I) : I is an independent set of G(cid:9).
Theorem 2.1. Let f ,...,f ∈ K[x] be m ≥ 2 polynomials such that 1,f ,...,f are K-linearly
1 m 1 m
independent. Then there exists constants C > 0 and a > 0 depending only on f ,...,f , such that
0 1 m
for any a > a and any T > a we have
0
C
(6) α¯(Cay(Km,S )) ≤ ,
T
T −a
where
S = S := (cid:8)±(f (s),...,f (s)) ∈ Km : s ∈ K, ea ≤ (cid:107)s(cid:107) ≤ eT(cid:9);
T a,T,f1,...,fm 1 m
here, e = e, the Euler number, when K = R and e = p when K = Q .
p
We always assume T and a are integers when K = Q .
p
Remark 2.2. By a theorem of Steinhaus, the difference set of a set of positive measure contains an
openneighborhoodofzero. Fromthisitfollowsimmediatelythatifzeroisinthetopologicalclosureof
S then α¯(Cay(Km,±S )) = 0. Hence in Theorem 2.1 we must pick a large enough so that 0 (cid:54)∈ S .
T T 0 T
In this case, one can show that the independence ratio is positive, as will be explained bellow.
Remark 2.3. Let f (0) = ··· = f (0) = 0, and assume that (f (t),...,f (t)) is non-zero for t (cid:54)= 0.
1 m 1 m
Denote S(cid:48) = {±(f (s),...,f (s)) : e−T ≤ s ≤ 1}. A careful analysis of the proof of Theorem 2.1
T 1 m
for K = R, shows that α¯(Cay(Km,S(cid:48) )) (cid:28) 1/T as T → ∞. This can be viewed as a quantitative
T
version of the phenomenon observed in Remark 2.2, accounting for the fact that the distance between
the set S(cid:48) and the origin is converging to zero. Moreover, when f (0) = ··· = f (0) = 0, the linear
T 1 m
independence condition in Theorem 2.1 can be dropped.
We remark that, when the Borel chromatic of G is finite, we have the following inequality
(7) χ (G)α¯(G) ≥ 1.
Bor
Fixing a large a > 0 and letting T → ∞, one can see that S is covered by a box of volume O(eNT)
T
for some N. From here one can easily see that χ (Cay(Km,S )) is bounded from above by CeNT
Bor T
for some C,N > 0 (independent of T). Thus by (7) we obtain Ce−NT ≤ α¯(Cay(Km,S )). By
T
combining (7) and Theorem 2.1, we obtain the following result.
Corollary 2.4. With the same notations as Theorem 2.1, we have:
(8) T −a (cid:28) χ (Cay(Km,S )) (cid:28) eNT,
Bor T
for some integer N.
Let us describe the key ideas of the proof of Theorem 2.1. First, we use a spectral upper bound
for the independence ratio of Cayley graph Cay(Km,S) established in [1] which is analogous to the
well-known Hoffman bound for finite graphs. The crucial step in the proof will consist of establishing
a uniform lower bound for the family of oscillatory real and p-adic integrals that arises as the Fourier
transform of a carefully chosen probability measures on S . The core of the proof is to use van
T
der Corput’s lemma to obtain a lower bound for these oscillatory integrals which is independent
of the frequency vector. It turns out that the frequency vectors that lie between two specific affine
hyperplanes have to be handled differently from the rest. Eventually, we will find a suitable partition
of the domain of integration into a uniformly bounded number of intervals (sphere in the p-adic case),
which enables us to apply van der Corput’s lemma and obtain a uniform lower bound for the Fourier
transform.
Theorem 2.1 can be reformulated in the following form, resembling Furstenberg-S´ark¨ozy’s theorem.
BOREL CHROMATIC NUMBER OF CAYLEY GRAPHS 5
Corollary 2.5 (High dimensional polynomial configurations). Let f ,...,f , a , C and e be as in
1 m 0
Theorem 2.1. For any T > a > a and any Borel set I ⊆ Km with
0
C
d¯(I) > ,
T −a
there exists s ∈ K with ea ≤ (cid:107)s(cid:107) ≤ eT and distinct x ,x ∈ I such that
1 2
x −x = (f (s),...,f (s)).
1 2 1 m
Remark 2.6. It is worth mentioning that even when f ,...,f have integer coefficients, the above
1 m
theoremisnotaconsequenceofFurstenbergandS´ark¨ozytheorem. ForinstancetakeI = {(x,y) ∈ R2 :
x2+y2 ∈ [0,1/4]+9N}, which has positive density in R2. Let f (x) = x and f (x) = x2+1. Observe
1 2
that I ∩Z2 has positive upper density in Z2 since (3N)2 ⊆ I ∩Z2. Due to arithmetic obstrauctions,
there are no x ,x ∈ I ∩Z2 and d ∈ Z such that x −x = (f (d),f (d)).
1 2 1 2 1 2
Corollary 2.7 (Polynomial configurations). Let f ,...,f , a , C and e be as in Theorem 2.1. For
1 m 0
any T > a > a and any Borel set I ⊆ K with
0
(cid:18) C (cid:19)1/m
d¯(I) > ,
T −a
there exists s ∈ K with ea ≤ (cid:107)s(cid:107) ≤ eT such that
{f (s),...,f (s)} ⊆ I −I.
1 m
Proof. Let I be as above. Then d¯(Im) > C/(T −a) and so by applying Corollary 2.5 we obtain the
result. (cid:3)
We now discuss the Borel chromatic number of Cayley graphs assigned to algebraic varieties.
Remark 2.8. By Steinhaus theorem, mentioned above, if zero is in the topological closure of S ⊆ Km
thenχ (Cay(Km,±S)) > ℵ . Conversely,if0 (cid:54)∈ S,thentheBorelchromaticnumberofCay(Km,±S)
Bor 0
is at most ℵ . Moreover, one can show that χ (Cay(Km,±S)) is finite when S is compact and 0 (cid:54)∈ S.
0 Bor
From Corollary 2.4 and Remark 2.8 we deduce the following qualitative result.
Corollary 2.9 (Infinite Borel chromatic number). Let f ,...,f be as in Theorem 2.1. Then for any
1 m
large enough a > 0 we have
(9) χ (Cay(Km,S) = ℵ ,
Bor 0
where
S = S = {±(f (s),...,f (s)) ∈ Km : s ∈ K, a ≤ (cid:107)s(cid:107)}.
a 1 m
The assumption that 1,f ,...,f are linearly independent is quite natural in this context. In fact
1 m
if 1,f ,...,f are linearly dependent, then S will be contained in an affine hyperplane in Km and one
1 m
can easily see that the Borel chromatic number of the Cayley graph associated to a hyperplane which
does not pass through the origin is finite.
The next theorem gives a partial answer to Question 1. Indeed, it shows that the Borel chromatic
numberoftheCayleygraphassignedtorealorp-adicvarietiesthatareparameterizedbypolynomials,
is infinite. Notice that by Remark 2.8, we need to remove a neighbourhood of the origin from the
algebraic set V in order to have a non-trivial answer for Question 1.
Theorem 2.10 (Multivariate polynomial configurations). Let F ,...,F ∈ K[x ,...,x ] be m ≥ 2
1 m 1 d
polynomials such that 1,F ,...,F are linearly independent. Then for any δ > 0 we have
1 m
(10) χ (Cay(Km,S )) = ℵ ,
Bor δ 0
where
S = {±(F (s),...,F (s)) ∈ Km : s ∈ Kd}\B .
δ 1 m δ
Proof. Suppose that the maximum degree of F ,...,F is (cid:96) − 1. Set n = (cid:96)i−1 for all 1 ≤ i ≤ d
1 m i
and substitute xi with tni. Thus the monomial xα11···xαdd will be substituted by th(α1,...,αd), where
h(α ,...,α ) = (cid:80)d α (cid:96)i−1. Notice that h is injective on the cube defined by 0 ≤ α ≤ (cid:96)−1. This
1 d i=1 i i
6 M.BARDESTANIANDK.MALLAHI-KARAI
clearly follows from the fact that α ,...,α are digits of h(α ,...,α ) when expressed in base (cid:96), and
1 d 1 d
hence are uniquely determined by h(α ,...,α ). Thus we obtain m ≥ 2 polynomials
1 d
f (t) := F (tn1,...,tnd),
i i
such that 1,f ,...,f are K-linearly independent. Then by applying Corollary 2.9 we obtain the
1 m
result. (cid:3)
Recall that the clique number of a graph G, denoted by ω(G), is the largest n for which G has a
subgraph isomorphic to the complete graph K . It is clear that ω(G) ≤ χ (G). Corollary 2.9 would
n Bor
be trivial if one could exhibit arbitrarily large cliques in G. However, we will show that there is an
algebraic obstruction to this, suggesting that the infinitude of the Borel chromatic number cannot be
proven by such local arguments. Obviously, it is enough to prove the nonexistence of large cliques
when K is replaced by an algebraic closure.
Theorem2.11. LetK beanalgebraicclosedfieldofcharacteristic0, andletV ⊆ Km beanirreducible
variety with dimV = 1 that is not an affine line. Then
(11) ω(Cay(Km,±V)) < ∞.
Another point that has to be addressed is the extent to which the polynomial nature of the map is
important for the above results to hold. We will show:
Theorem 2.12. There exists a real analytic curve f : (a,b) → R2 such that the image of f does
not lie between any two parallel lines and the Borel chromatic number of the Cayley graph of R2 with
respect to the image of f is finite.
As indicated above, proving statements analogous to Theorem 2.10 for the ordinary chromatic
number seems to be quite difficult. This is essentially due to the fact that proving such an statement
is equivalent to finding (or proving the existence of) finite subgraphs of arbitrarily large chromatic
number inside the Cayley graphs of Kn. In the last section of this paper, we will prove a general
theorem that highlights this point by showing the question of determining the chromatic number over
C is indeed algebraic in nature. This theorem (and the techniques used in its proof) are independent
of the rest of the paper, and in solely included for the sake of comparison.
Let V be an algebraic set defined over Q given by polynomials F ∈ Q[x ,...,x ] for 1 ≤ i ≤ n.
i 1 m
Without loss of generality, we can assume that F ∈ Z[x ,...,x ]. If R is any ring of characteristic
i 1 m
zero, we can define
V(R) = {(x ,...,x ) ∈ Rm : F (x ,...,x ) = 0, 1 ≤ i ≤ n}.
1 m i 1 m
For p prime, we will denote by Z the ring of p-adic integers.
p
Theorem 2.13. Let V be an irreducible variety defined over Q. Then
(12) χ(Cay(Cm,±V(C))) = supχ(Cay(Zm,±V(Z ))),
p p
p
if one of the two sides is finite. The sup is taken over all primes p.
The proof of Theorem 2.13 relies on a theorem of de Bruijn and Erd˝os which relates the chromatic
number of a graph to its finite subgraphs and an embedding theorem of Cassels.
3. Preliminaries
We begin by recalling some basic facts on Borel chromatic number. Let us first introduce some
notation. Throughout the paper we let S ⊂ Km denote a symmetric set, i.e, −S = S, and 0 (cid:54)∈ S.
We define the Cayley graph G := Cay(Km,S) to be the graph with the vertex set Km, in which two
vertices x,y ∈ Km are adjacent if and only if x−y ∈ S. Recall that I ⊆ Km is called an independent
set of G if (I −I)∩S = ∅.
Lemma 3.1. With the above notations, χ (G) ≤ ℵ if and only if 0 (cid:54)∈ S¯.
Bor 0
BOREL CHROMATIC NUMBER OF CAYLEY GRAPHS 7
Proof. First assume that 0 (cid:54)∈ S¯. Choose δ > 0 such that B ∩S = ∅. This, in particular, implies that
2δ
Bδ isanindependentsetforG. NotethatsinceQm isdenseinKm, wehaveKm = ∪v∈Qn(v+Bδ). This
countable cover can then be easily transformed further into a disjoint countable cover of independent
Borel (in fact, locally closed) sets, yielding the desirable coloring.
Now assume χ (G) ≤ ℵ . From countable additivity, it follows that there exists a color class I
Bor 0
with positive Lebesge measure. Using a theorem of Steinhaus (whose proof works word-for-word in
the p-adic case), there exists δ > 0 such that B ⊆ (I −I). Since I is independent, it follows that
1 δ1
S ∩B = ∅, and so 0 (cid:54)∈ S¯. (cid:3)
δ1
If, furthermore, S is a bounded Borel set, then the Borel chromatic number of G is finite.
Lemma 3.2. Let S be a bounded Borel set such that 0 (cid:54)∈ S¯. Then χ (G) is finite.
Bor
Proof. See [3, Lemma 3.3]. (cid:3)
Since invertible linear maps preserve the additive structure, the following lemma is immediate.
Lemma 3.3. Let B ∈ GL (K) be an invertible matrix. Then for any symmetric set S ⊆ Km we have
m
χ (Cay(Km,S)) = χ (Cay(Km,B(S))).
Bor Bor
Moreover, the same is true for the ordinary chromatic number.
4. A spectral bound for Independence ratio
Let G = (V,E) be a finite regular graph with n vertices and the adjacency matrix A. Let
λ ≥ λ ≥ ··· ≥ λ ,
0 1 n−1
denote the spectrum of A and assume that I ⊆ V is an independent set of G. The celebrated Hoffman
bound states that
|I| λ λ
n−1 min
(13) ≤ − = − .
n λ −λ λ −λ
0 n−1 max min
This yields the following upper bound for the chromatic number:
λ −λ
max min
(14) χ(G) ≥ − .
λ
min
It is a reinterpretation of this inequality that is the key to the generalization we will need later.
Denote by L2(V) the Hilbert space of all complex-valued functions on V, equipped with the inner
product (cid:104)f,g(cid:105) := (cid:80) f(v)g(v). The adjacency operator A : L2(V) → L2(V), defined by Af(v) =
v∈V
(cid:80)
f(w), is easily seen to be self-adjoint. Further, one can see that its numerical range defined by
vw∈E
W(A) := {(cid:104)Af,f(cid:105) : (cid:107)f(cid:107) = 1},
2
is equal to [λ ,λ ]. It is a routine verification that a subset I ⊆ V is independent in the graph G
n−1 0
if and only if for all f ∈ L2(V) supported on I, one has (cid:104)Af,f(cid:105) = 0. This novel interpretation of
independent sets is used in [1] to prove an analog of the Hoffman bound for certain Cayley graphs
of the Euclidean additive group Rn. As we will need to work in a slightly more general framework,
it will be useful to briefly review some of the key points of [1]; for details, we refer the reader to the
original paper [1].
Abstractly, let (V,Σ,µ) be a probability space, consisting of a set V, a σ-algebra Σ on V, and
a probability measure µ, and consider the Hilbert space L2(V) of square integrable functions with
respect to the inner product:
(cid:90)
(cid:104)f,g(cid:105) = f(x)g(x)dµ(x).
V
For a bounded and self-adjoint operator A : L2(V) → L2(V), one can show that the numerical range
of A defined by
W(A) = {(cid:104)Af,f(cid:105) : (cid:107)f(cid:107) = 1},
2
is an interval in R. We denote the endpoints of W(A) by
m(A) := inf{(cid:104)Af,f(cid:105) : (cid:107)f(cid:107) = 1}, M(A) := sup{(cid:104)Af,f(cid:105) : (cid:107)f(cid:107) = 1}.
2 2
8 M.BARDESTANIANDK.MALLAHI-KARAI
Definition 4.1. Let A : L2(V) → L2(V) be a bounded, self-adjoint operator. A measurable set I ⊆ V
is called an independent set for A if (cid:104)Af,f(cid:105) = 0 for each f ∈ L2(V) which vanishes almost
everywhere outside of I. Moreover the chromatic number of A, denoted by χ(A), equals the least
number k such that one can partition V into k independent sets for A.
The independence ratio of A is defined by
(15) α¯(A) := sup{µ(I) : I is an independent set of A }.
The following theorem—which can be obtained by a clever modification of the proof of Hoffmann’s
bound presented by Bollob´as [7, Chapter VIII.2]—recovers (13) when A is the adjacency matrix of a
finite regular graph.
Theorem 4.2. Let (V,Σ,µ) be a probability space and let A : L2(V) → L2(V) be a nonzero, bounded,
self-adjoint operator. Fix a real number R and set ε = (cid:107)A1−R1(cid:107) , where 1 = 1 is the characteristic
2 V
function of V. Suppose there exists a set I ⊆ V with µ(I) > 0 which is independent for A. Then, if
R−m(A)−(cid:15) > 0, we have
−m(A)+2(cid:15)
(16) α¯(A) ≤ .
R−m(A)−(cid:15)
Proof. See [1, Theorem 2.2]. (cid:3)
Moreover,Bachoc,DeCortede,OliveiraandVallentinproved[1,Theorem2.3]thefollowingtheorem
which is an analogue of (14)
Theorem 4.3. Let A : L2(V) → L2(V) be a nonzero, bounded and self-adjoint operator. Moreover
assume that χ(A) < ∞. Then
M(A)
(17) χ(A) ≥ 1− .
m(A)
Remark 4.4. Similar to the Hoffman bound, when A (cid:54)= 0 and χ(A) < ∞ the proof of the above
theorem shows that m(A) < 0 and M(A) > 0.
We now apply these theorems to the Borel chromatic number and indepndence ratio of Cayley
graphs with the vertex set Km. All that follows is parallel to [1], where the case K = R is dealt with;
in fact the arguments in [1] can be easily seen to work also in the p-adic case. For the convenience of
the reader we will briefly sketch the key points.
Let us denote by dx a Haar measure on K. When K = Q , we assume that it is normalized such
p
that (cid:82) dx = 1. We will also write dx for the product (Haar) measure dx ···dx on Km. For a
Z 1 m
Borel sept E ⊆ Km we use |E| = (cid:82) dx to denote the measure of E. As before, S ⊆ Km is a symmetric
E
(i.e., −S = S), Borel measurable set (with respect to the σ-algebra generated by the open sets), which
does not contain the origin in its closure.
Throughout this section, we let µ be a probability Borel measure on Km supported in S.
Therefore µ is a Radon measure [15, Theorem 7.8]. We will also assume that µ is symmetric, i.e.,
µ(−A) = µ(A) holds for all Borel measurable sets A. Define the bounded, self-adjoint operator A
µ
by
A : L2(Km) → L2(Km), f (cid:55)→ f ∗µ,
µ
where
(cid:90)
(18) f ∗µ(x) = f(x−y)dµ(y).
Km
From now on, assume that S is also bounded. By Lemma 3.2, the Borel chromatic number of
Cay(Km,S) is finite. Since Supp(µ) ⊆ S, any independent set I for Cay(Km,S) is also an inde-
pendent set for A . Therefore by Theorem 4.3 we have
µ
M(A )
(19) 1− µ ≤ χ(A ) ≤ χ (Cay(Km,S)).
m(A ) µ Bor
µ
BOREL CHROMATIC NUMBER OF CAYLEY GRAPHS 9
The numerical range of A can now be determined using Fourier analysis. Below, we will review a
µ
number of basic properties of the Fourier transform over R and Q . For details we refer the reader
p
to [25, 30]. For K = R, we will use the character
ψ : R → C∗, ψ(x) = exp(2πix).
When K = Qp, for x ∈ Qp, we write nx for the smallest non-negative integer such that pnxx ∈ Zp. Let
rx ∈ Z be such that rx ≡ pnxx (mod pnx). It is well-known that the following map (called the Tate
character)
(cid:18) (cid:19)
2πir
(20) ψ : Q → C∗, ψ(x) = exp x ,
p pnx
is a non-trivial character of (Q ,+) with the kernel Z .
p p
The Fourier transforms of f ∈ L1(Km) and µ are respectively defined by the integrals
(cid:90) (cid:90)
f(cid:98)(u) = ψ(x·u)f(x)dx, µ(cid:98)(u) = ψ(x·u)dµ(x).
Km Km
Here x · u is the standard bilinear form on Km, and ψ is the complex conjugate of ψ. We remark
that µ is a real-valued function since µ is symmetric. By Plancherel’s theorem the Fourier transform
(cid:98)
extends to an isometry on L2(Km) and thus for any f ∈ L2(Km) we have
(cid:91)
(21) (cid:104)Aµf,f(cid:105) = (cid:104)A(cid:100)µf,f(cid:98)(cid:105) = (cid:104)f ∗µ,f(cid:98)(cid:105) = (cid:104)µ(cid:98)f(cid:98),f(cid:98)(cid:105).
Lemma 4.5. Let µ be a symmetric probability Borel measure on Km with its support contained in S.
Then the numerical range of A is given by
µ
m(A ) = inf µ(u), 1 = M(A ) = sup µ(u).
µ (cid:98) µ (cid:98)
u∈Km u∈Km
Proof. From (21) combined with the fact that the Fourier transform on L2(Km) is an isometry, we
deduce that the numerical range of the operator A is the same as the numerical range of the multipli-
µ
cation operator g (cid:55)→ µg. Since µ is a symmetric probability Borel measure, µ is a bounded continuous
(cid:98) (cid:98)
real-valued function. Now let g ∈ L2(Km) with (cid:107)g(cid:107) = 1. Evidently
2
(cid:90)
(cid:104)µg,g(cid:105) = µ(x)|g(x)|2dx ≥ inf µ(u),
(cid:98) (cid:98) (cid:98)
Km u∈Km
and so m(Aµ) ≥ infu∈Kmµ(cid:98)(u). Now let ε > 0 and pick x0 ∈ Km with µ(cid:98)(x0) ≤ infu∈Kmµ(cid:98)(u)+ε. Let
B = B (x ) be the ball of radius δ centered at x . Since µ is continuous, we conclude that
δ δ 0 0 (cid:98)
(cid:12) (cid:12)2
(cid:90) (cid:12)1 (x)(cid:12) 1 (cid:90)
lim µ(x)(cid:12) Bδ (cid:12) dx = lim µ(x)dx = µ(x ),
δ→0 Km (cid:98) (cid:12)(cid:12)(cid:112)|Bδ|(cid:12)(cid:12) δ→0 |Bδ| Bδ (cid:98) (cid:98) 0
where 1 is the characteristic function of B . Hence
Bδ δ
m(A ) ≤ µ(x ) ≤ inf µ(u)+ε,
µ (cid:98) 0 (cid:98)
u∈Km
and therefore we have m(Aµ) = infu∈Kmµ(cid:98)(u). Similarly, since µ is a probability measure, we have
M(A ) = sup µ(u) = 1. (cid:3)
µ u∈Km (cid:98)
We can now summarize these in the following theorem.
Theorem 4.6. Let K be either R or a p-adic field Q and let S be a bounded symmetric Borel
p
measurable subset of Km which does not contain the origin in its closure. Then for any symmetric,
probability Borel measure µ on Km with its support contained in S we have
1
(22) 1− ≤ χ (Cay(Km,S)).
infu∈Kmµ(cid:98)(u) Bor
Let us now consider the independence ratio of Cay(Km,S). For r > 0, let B ⊆ Km be the ball of
r
radius r centered at the origin, and normalize the induced measure on B . Define
r
Ar : L2(B ) → L2(B ),
µ r r
by
Ar(f) := (cid:0)A f¯(cid:1)| ,
µ µ Br
10 M.BARDESTANIANDK.MALLAHI-KARAI
where f¯is the extension of f to Km defined to be zero on Km \B . By abuse of notation, we will
r
continue to write f for f¯. As mentioned before, any independent set I of Cay(Km,S) is also an
independent set of A . Hence I ∩B is an independent set for the operator Ar. From the definition
µ r µ
of the independence ratio of an operator, we obtain the following density bound:
|B ∩I|
r ≤ α¯(Ar).
|B | µ
r
This implies that
(23) α¯(Cay(Km,S)) ≤ limsupα¯(Ar).
µ
r→∞
For a given r > 0, define
(24) R = R(r) = (cid:104)Ar1 ,1 (cid:105) , (cid:15) = (cid:15)(r) = (cid:107)Ar1 −R(r)1 (cid:107) ,
µ Br Br L2(Br) µ Br Br L2(Br)
where 1 is the characteristic function of B .
Br r
Lemma 4.7. With the above notation, we have
(25) lim m(Ar) = m(A), lim R(r) = 1, lim (cid:15)(r) = 0.
µ
r→∞ r→∞ r→∞
Proof. Let f ∈ L2(Km). Define fr := f . Then
|Br
(cid:104)A fr,fr(cid:105) (cid:104)fr,fr(cid:105)
(cid:104)Arfr,fr(cid:105) = µ L2(Km) ≥ m(A ) L2(Km) = m(A )(cid:104)fr,fr(cid:105) .
µ L2(Br) |B | µ |B | µ L2(Br)
r r
Hence m(Ar) ≥ m(A ). Conversely, as r → ∞, we have
µ µ
(cid:104)Arfr,fr(cid:105) (cid:104)A fr,fr(cid:105) (cid:104)A f,f(cid:105)
m(Ar) ≤ µ L2(Br) = µ L2(Km) −→ µ L2(Km),
µ (cid:104)fr,fr(cid:105) (cid:104)fr,fr(cid:105) (cid:104)f,f(cid:105)
L2(Br) L2(Km) L2(Km)
since fr → f in L2(Km). This shows that lim m(Ar) = m(A ).
r→∞ µ µ
Recall that µ is a probability measure with Supp(µ) ⊆ S. Thus for every x ∈ B we have
r
(cid:90)
(26) Ar1 (x) = 1 (x−y)dµ(y).
µ Br Br
S
S is bounded, and so we assume that S ⊆ B for some d > 0. For K = Q we get the desired limits
d p
since for any r > d, by the ultrametric metric inequality, we obtain Ar1 (x) = 1. Therefore assume
µ Br
that K = R. Let r > d and x ∈ B . Then from (26) we deduce that Ar1 (x) = 1 for all x ∈ B .
r−d µ Br r−d
Hence
(cid:90)
1
lim R(r) = lim Ar1 (x)dx = 1.
r→∞ r→∞ |Br| Br µ Br
Finally
(cid:90)
1 |B \B |
|Ar1 (x)−1|2dx ≤ 2 r r−d → 0, r → ∞.
|B | µ Br |B |
r Br r
Then by the triangular inequality we deduce that lim (cid:15)(r) = 0. (cid:3)
r→∞
Therefore by applying Theorem 4.2, Lemma 4.5, Lemma 4.7 and (23) we obtain the following
theorem:
Theorem 4.8. Let K be either R or a p-adic field Q and let S be a bounded, symmetric Borel subset
p
of Km which does not contain the origin in its closure. Then for any symmetric, probability Borel
measure µ on Km with its support contained in S we have
(27) α¯(Cay(Km,S)) ≤ − infu∈Kmµ(cid:98)(u) .
1−infu∈Kmµ(cid:98)(u)