The square root law and structure of finite rings
aa r X i v : . [ m a t h . N T ] M a y THE SQUARE ROOT LAW AND STRUCTURE OF FINITERINGS
A. IOSEVICH, B. MURPHY, AND J. PAKIANATHAN
Abstract.
Let R be a finite ring and define the hyperbola H = { ( x, y ) ∈ R × R : xy = 1 } . Suppose that for a sequence of finite odd order ringsof size tending to infinity, the following “square root law” bound holdswith a constant C > χ on R : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( x,y ) ∈ H χ ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C p | H | . Then, with a finite number of exceptions, those rings are fields.For rings of even order we show that there are other infinite familiesgiven by Boolean rings and Boolean twists which satisfy this square-rootlaw behavior. We classify the extremal rings, those for which the lefthand side of the expression above satisfies the worst possible estimate.We also describe applications of our results to problems in graph theoryand geometric combinatorics.These results provide a quantitative connection between the squareroot law in number theory, Salem sets, Kloosterman sums, geometriccombinatorics, and the arithmetic structure of the underlying rings. Introduction
The square root law is a ubiquitous concept in modern mathematics.Roughly speaking, it says that (cid:12)(cid:12)(cid:12)X oscillating terms of modulus (cid:12)(cid:12)(cid:12) ≤ C p terms. Many classical problems and open conjectures can be related to the squareroot law. For example, the Riemann hypothesis can be restated [16] in termsof the square root law applied to the exponential sum X
Even if we were to only stick to famous problems, the list of situationswhere the square root law comes into play is very large. An interestedreader can take a look at a very informative survey by Barry Mazur [14]where several aspects of this concept are exposed. The manifestation of thesquare root law that is most relevant to us is Deligne’s proof [4, 5] of theRiemann hypothesis for finite fields. See also [11] for a very nice survey ofthe problem. One of the key aspects of this theory is obtaining sharp boundsfor Kloosterman type sums [17], in particular the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X s ∈ F ∗ q χ ( as + bs − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ q, where F q is the finite field with q elements, ( a, b ) ∈ F q \{ (0 , } , F ∗ q is thefield’s multiplicative group and χ is a non-trivial additive character on F q .The square root law is discussed in this paper in the context of Salemsets . Definition 1.1.
Let { R i } ∞ i =1 denote a set of finite rings such that | R i | → ∞ as i → ∞ . Let { S i } ∞ i =1 denote the collection of sets such that S i ⊂ R di , the d -dimensional module over R i . Let b S i ( γ i ) denote the Fourier transform of S i , viewed as the characteristic function of the set S i .(i) Suppose that for every ǫ > C ǫ >
0, independent of i ,such that | b S i ( γ i ) | ≤ C ǫ | R i | − d | S i | + ǫ for every non-zero γ i ∈ Γ i . Then we say that { S i } ∞ i =1 is Salem withrespect to { R di } ∞ i =1 .(ii) Suppose that there exists C > | b S i ( γ i ) | ≤ C | R i | − d | S i | for every non-zero γ i ∈ Γ i .Then we say that { S i } ∞ i =1 is purely C -Salem with respect to { R di } ∞ i =1 .The sphere provides us with a way to construct examples of Salem setsin a discrete setting. Let F q denote the finite field with q elements. Let S t = n x ∈ F dq : || x || ≡ x + · · · + x d = t o , where t is a unit in F q . It is well-known (see for example [6]) that | b S t ( γ ) | ≤ q − d q d − if γ is non-zero. Since | S t | = q d − + lower order terms, we instantly recoverthe Salem property with ǫ = 0 for any sequence of fields. The proof in Here and throughout, if S is a finite set, | S | denotes the number of elements of S . See section 2 for the definition of the Fourier transform.
ALEM SETS IN MODULES OVER FINITE RINGS 3 [6] also shows that the hyperboloid is a pure Salem set for any sequence offields, and a Gauss sum estimate shows the same for the paraboloid.We have already seen that some explicitly defined sets such as the sphere,paraboloid and the hyperboloid are all Salem sets over finite fields F q .Doesthis phenomenon persist over more general rings? The main thrust of thispaper is that the answer is, in general, no, at least for odd order rings. In thecase of even rings, the main culprits are large Boolean rings and we classifythe set of exceptions below. Theorem 1.2.
Let { R i } ∞ i =1 denote a sequence of odd order finite rings with | R i | → ∞ . Let H i = { x ∈ R i : x x = 1 } denote the hyperbola with respect to the ring R i .Suppose that { H i } ∞ i =1 is purely Salem with respect to { R i } ∞ i =1 . Then thereexists i such that for all i > i R i is a field. Our proof examines rings of general (odd or even) order also and ourresults show that if one restricts to finite rings which have no Z / Z -factors intheir semisimple decomposition, or even a bounded number of such factors,then Theorem 1.2 still holds in the sense that if the hyperbola is purelySalem with respect to such a sequence rings, then all but finitely many ofthese rings are fields.In the presence of an unlimited number of Z / Z factors, we establishother sequences which have a purely Salem hyperbola. Specifically, we showthat if R n = Z / Z × · · · × Z / Z is the Boolean ring of order 2 n then the sequence of these rings has a purelySalem hyperbola. More generally if R is any fixed finite ring, the sequence S n = R × R n has a purely Salem hyperbola.Theorem 5.14 on page 14 of [1] shows that a random construction yieldsSalem sets with respect to any sequence of rings R i of size tending to infin-ity. However such constructions result in a logarithmic loss, which meansthat the resulting sequence is Salem, but not purely Salem (see Definition1.1 above). Salem sets share many properties with random sets, so it isinteresting when a specific set of geometric and arithmetic importance likethe hyperbola, sphere, or parabola exhibits Salem set behavior.The Fourier coefficients of the hyperbola can be interpreted as generalizedKloosterman sums in the resulting rings as explained in Section 2. Thesesums are an important class of exponential sums with numerous numbertheoretic applications. In the process of proving Theorem 1.2 we introducethe concept of the Kloosterman-Salem number of a finite ring R , denoted by C R , which measures quantitatively how well the “square-root law” holds forKloosterman sums over that ring. More precisely, the Kloosterman-Salem A. IOSEVICH, B. MURPHY, AND J. PAKIANATHAN number is the smallest positive number C such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ R ∗ χ m ( x ) χ n ( x − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C p | R ∗ | for all ( m, n ) ∈ R \ { (0 , } .The larger this number, the weaker the form of the resulting square-rootlaw. In the course of proving Theorem 1.2, we show that for any threshold α ∈ (0 , ∞ ), only a finite number of non-field odd order finite rings haveKloosterman-Salem number C R < α . On the other hand, results of Weil,Deligne, and Nicholas Katz show finite fields F have Kloosterman-Salemnumber asymptotic to 2 as | F | → ∞ . For details please refer to Section 2below.The following theorem, interesting in its own right, summarizes our quan-titative results on the Kloosterman-Salem numbers. Theorem 1.3.
Let α ∈ (0 , ∞ ) be a threshold. Then • If α < , only a finite number of odd order finite rings have Kloosterman-Salem number C R with C R ≤ α . • If α > , only a finite number of odd order finite rings which are notfields have C R ≤ α . All but at most a finite number of finite fieldshave C R ≤ α . • lim | F |→∞ C F = 2 where the limit is taken over any sequence of finitefields. • Every finite ring has ≤ C R ≤ p | R ∗ | . The rings with C R = 1 areexactly the finite Boolean rings. • The finite rings with C R = p | R ∗ | are called extremal rings. A finitefield is extremal if and only if it has order , or . For any finitering R , S = R × B , where B is a nontrivial Boolean ring, has S anextremal ring. • Every finite non-Boolean ring has C R ≥ √ . These results are illustrated in Figure 1.1.1.
A graph theoretic viewpoint.
The methods of this paper can beused to yield some light on Erd˝os-type problems in geometric combinatoricsand related graph theoretic questions. This subsection is dedicated to a briefdiscussion of these topics.Let R be a finite associative ring with a unit and define a graph, calledthe hyperbola graph of R , with vertices being the elements of R and twovertices x and y are connected by an edge if x − y ∈ H , where H is ahyperbola defined above. ALEM SETS IN MODULES OVER FINITE RINGS 5 C F N Booleanrings √ F C F | F |→∞ √ α finite Figure 1.
Kloosterman-Salem numbers
Definition 1.4.
The clique number of a graph is the number of vertices inthe largest complete subgraph. The independence number of a graph is thenumber of vertices in the largest edgeless subgraph. The chromatic numberof a graph is the smallest number of distinct colors for the vertices such thatif two vertices are connected by an edge, they are of different color.
Definition 1.5.
The spectrum of a graph is the collection of eigenvalues ofits adjacency matrix. A graph is regular if all its vertices have the samedegree d and for such a graph d is the largest eigenvalue of the adjacencymatrix, and indeed the spectrum lies in the interval [ − d, d ]. The value − d is in the spectrum of a d -regular graph if and only if the graph is bipartite.For a non-bipartite graph, the spectral gap is the size of the gap between thelargest and 2nd largest eigenvalue. The spectral gap is defined to be zero ifand only if d has multiplicity ≥ Theorem 1.6.
Let R be an associative finite ring with a unit. Let C R denote the Kloosterman-Salem number of R . Then: • The hyperbola graph is a regular graph of degree d = | R ∗ | and it isconnected and not bipartite if and only if C R < p | R ∗ | i.e., the ring isnot extremal. • The spectrum of the hyperbola graph consists exactly of | R | times theFourier coefficients of the hyperbola’s characteristic function. • For a non-extremal ring, the spectral gap of the hyperbola graph is | R ∗ | − C R p | R ∗ | . • In the case that the hyperbola graph is connected and not bipartite (i.e. R is non-extremal), a random walk on the graph is mixing i.e. con-verges to the uniform distribution at a rate determined by the spectralgap. More precisely for every starting node i , the probability p tij thatafter t steps in a uniform random walk on the hyperbola graph, that A. IOSEVICH, B. MURPHY, AND J. PAKIANATHAN we end up at vertex j satisfies: (cid:12)(cid:12)(cid:12)(cid:12) p tij − | R | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C R p | R ∗ | ! t . • The independence number of the hyperbola graph of R is at most C R | R | | R ∗ | − . • The chromatic number of the hyperbola graph of R is at least | R ∗ | C R . • In particular, if R is a finite field of order q , then the chromatic numberof the hyperbola graph is at least . √ q − . As an aside, note that any sequence of distinct finite rings with C R = o ( p | R ∗ | ) yields a sequence of hyperboloid graphs with chromatic numbertending to infinity. 2. Proof of Theorem 1.2
Basic setup.
Let R be a finite ring, which is associative with identitybut not necessarily commutative. We view the hyperbola as H = × − (1)where × : R × R → R is the ring multiplication. Thus H = { ( x, y ) ∈ R : xy = 1 } . Clearly | H | = | R ∗ | where R ∗ is the unit group of R .We now identify R with its Pontryagin dual as its underlying additivegroup is finite abelian. We then identify the Pontryagin dual of R withitself accordingly. The Haar measure is the counting measure normalizedso that the entire space has measure 1. With this notation, the Fouriertransform of the hyperbola H ⊆ R is given byˆ H ( m ) = 1 | R | X ( x,y ) ∈ R H ( x, y ) χ m ( − ( x, y )) = 1 | R | X x ∈ R ∗ χ m ( − x ) χ m (cid:18) − x (cid:19) , where χ m is the character in the dual group corresponding to m ∈ R underthe identification of R with its dual, and m = ( m , m ) ∈ R .When the underlying abelian group of R is cyclic, we can write χ m ( x ) = χ ( mx ) where χ is a fixed non-trivial character and this becomes the well-known Kloosterman sumˆ H ( m ) = 1 | R | X x ∈ R ∗ χ (cid:16) − m x − m x (cid:17) for m = ( m , m ) ∈ R .When R is a finite field, it is well-known [17] that(2.1) | ˆ H ( m ) | ≤ | R | − p | R | for m = 0 . ALEM SETS IN MODULES OVER FINITE RINGS 7
Thus the hyperbola is a pure C -Salem set for some C ≤ p | R | / | R ∗ | = 2 q − | R | when R is a finite field. Definition 2.1 (Kloosterman-Salem number) . The
Kloosterman-Salem num-ber of R is the infimum of numbers C > | ˆ H ( m ) | ≤ C | R | − | R ∗ | for all m = (0 , C R .The Kloosterman-Salem number is clearly finite and non-negative for anyfinite ring. Note that C R = | R | p | R ∗ | max m =(0 , | ˆ H ( m ) | . If the Kloosterman-Salem number of R is C , we say that R is a pure C -Salemset .The vertical equidistribution of Kloosterman sums over finite fields, es-tablished by Nicholas Katz [8], implies that the constant 2 in Weil’s boundis asymptotically sharp: lim | F |→∞ C F = 2 , where the limit is taken over finite fields F . We will establish that for anythreshold 0 < α < ∞ , there are only finitely many finite rings of odd orderaside from fields with C R < α . This means for thresholds 0 < α < C R < α whereas forthresholds 2 < α < ∞ almost all fields have C R < α whereas only finitelymany non field, odd order finite rings have C R < α .In particular this means that any sequence { R n } of distinct finite oddorder rings with Kloosterman-Salem number uniformly bounded, are even-tually fields, in the sense that there exists N >
0, such that for n ≥ N , R n is a field. This is the essence of Theorem 1.2.2.2. A geometric criterion.
An important set that encodes the connec-tion between addition + and multiplication × in the ring R is given by N ( R ) = ( × ◦ − ) − (1)= ( − ) − ( H )= { ( x, y ) ∈ R × R : x − y ∈ H } = { ( x, y ) ∈ R × R : ( x − y )( x − y ) = 1 } where − : R × R → R is subtraction and × : R × R → R is multiplication. A. IOSEVICH, B. MURPHY, AND J. PAKIANATHAN
Similarly, we define N ( E ) = { ( x, y ) ∈ E × E : x − y ∈ H } = { ( x, y ) ∈ E × E : ( x − y )( x − y ) = 1 } and let n ( E ) = | N ( E ) | .The next result relates the Kloosterman-Salem number to the size of theset N ( E ). Theorem 2.2.
Let R be a finite ring with Kloosterman-Salem number C .Then any set E ⊂ R with | E | > C | R | | R ∗ | has n ( E ) > . More precisely, thereexist e , e ∈ E such that e − e ∈ H .Proof. Let q = | R | . Then we have: n ( E ) = |{ ( x, y ) ∈ E × E : x − y ∈ H }| = X x,y E ( x ) E ( y ) H ( x − y )= q X m | ˆ E ( m ) | ˆ H ( m )= q X m =0 | ˆ E ( m ) | ˆ H ( m ) + | E | | R ∗ | q = D ( E ) + | E | | R ∗ | q where D ( E ) = P m =0 | ˆ E ( m ) | ˆ H ( m ) is called the discrepancy of the set E relative to the hyperbola H .As H is a pure C -Salem set of size | R ∗ | , we have | D ( E ) | ≤ X m | ˆ E ( m ) | ! · Cq − | R ∗ | = q | E | Cq − | R ∗ | where the last step follows by the Plancherel theorem. Thus | D ( E ) | ≤ C | E || R ∗ | . As n ( E ) = D ( E ) + | E | | R ∗ | q as long as | D ( E ) | < | E | | R ∗ | q , we will have n ( E ) >
0. This is certainly the case when C | E || R ∗ | < | E | | R ∗ | q which happens when | E | > Cq | R ∗ | . Thus Theorem 2.2 is proven. (cid:3) ALEM SETS IN MODULES OVER FINITE RINGS 9
Bound on the size of ideals.
The sum-product formulation encap-sulated in Theorem 2.2 leads directly to a bound on the size of proper idealsof R in terms of the Kloosterman-Salem number C . Theorem 2.3.
Let R be a finite ring with unit, with Kloosterman-Salemnumber C . Then any proper left (or right) ideal I of R has | I | ≤ C | R || R ∗ | . Remark . As every ring (with unit) has at least the zero proper ideal,this in particular implies
C >
Proof.
We prove the theorem for proper left ideals. The proof for right idealsis similar. If I is a proper left ideal then E = R × I ⊆ R has n ( E ) = 0 as itis impossible to solve the equation ( x − y )( x − y ) = 1 since x − y ∈ I .Thus | E | = | R || I | ≤ C | R | | R ∗ | by Theorem 2.2. This completes the proof ofTheorem 2.3. (cid:3) We will see that the bound in Theorem 2.3 is sharp in the sense that forany fixed 0 < C < ∞ , only a limited class of finite rings satisfy it.2.4. Structure of finite rings.
Finite rings have a well studied structure,which we record here:
Proposition 2.5.
Let R be a finite ring with unit.(1) R contains a unique two-sided maximal ideal J , called the Jacobsonradical , such that
R/J is a semi-simple ring.(2) The semi-simple quotient
R/J is isomorphic to a product of matrixrings over finite fields.
This was one of the first complete classification theorems in algebra. Wewill sketch a proof, with references to Lang’s
Algebra [12], where the readercan find the details.
Proof. If R is a finite ring with Jacobson radical J then R/J is finite andsemisimple. Since semisimple rings are direct products of simple rings(Chapter XVII, Theorem 4.4), it follows that
R/J is a finite product ofsimple rings. Finally, every finite simple ring is a matrix ring over a finitefield.This last fact follows from two famous theorems. First, as finite rings areArtinian, the Artin-Wedderburn theorem shows that finite simple rings areisomorphic to Mat n ( D ), the n × n matrix ring over a finite division ring D .It follows immediately from Wedderburn’s theorem, which states that finitedivision rings are fields, that D is a finite field. (cid:3) Now we will fix notation. Let R = Mat n ( F ) where F is a finite field.Then | R | = | F | n and | R ∗ | = | GL n ( F ) | = | F | n (cid:18) − | F | (cid:19) (cid:18) − | F | (cid:19) . . . (cid:18) − | F | n (cid:19) . Thus | R ∗ || R | = φ R = φ ( n, | F | ) = (cid:18) − | F | (cid:19) (cid:18) − | F | (cid:19) . . . (cid:18) − | F | n (cid:19) . It will be convenient to have a uniform lower bound on φ ( n, | F | ). Lemma 2.6.
For all n ≥ and all finite fields F , we have φ ( n, | F | ) ≥ / .Proof. Since φ ( n, | F | ) = (1 − | F | )(1 − | F | ) . . . (1 − | F | n ) is increasing in | F | ,the general case follows from the case | F | = 2. As φ ( n,
2) is monotonicallydecreasing as a function of n , it then suffices to establish that the infiniteproduct α = (cid:18) − (cid:19) (cid:18) − (cid:19) . . . (cid:18) − n (cid:19) . . . is bounded below by . A priori, by the monotone convergence theorem, α exists in the interval [0 , − x ) we get: − log( α ) = ∞ X n =1 ∞ X k =1 k nk Exchanging the order of summation and summing the geometric series as afunction of n , we get − log( α ) = ∞ X k =1 1 k k − k = ∞ X k =1 k (2 k − ≤ ∞ X k =1 k (cid:18) (cid:19) k − = − (cid:18) (cid:19) Thus log( α ) ≤ log(4) yielding α ≥ as desired. (cid:3) If R is a finite semisimple ring, then by Proposition 2.5(2.2) R = Mat n ( F ) × · · · × Mat n k ( F k ) , where F , . . . , F k are finite fields labelled so that | F | n ≤ | F | n ≤ · · · ≤| F k | n k . If R is written as in (2.2), we say that R has k semisimple factors .Note that R ∗ = GL n ( F ) × · · · × GL n k ( F k ) , and | R ∗ | = | R | φ ( n , | F | ) · · · φ ( n k , | F k | ) . Finally, if R is a finite ring with Jacobson radical J then by Proposition2.5, we have the short exact sequence of rings and ideals:0 → J → R → Mat n ( F ) × · · · × Mat n k ( F k ) → , ALEM SETS IN MODULES OVER FINITE RINGS 11 where again F , . . . , F k are finite fields labelled so that | F | n ≤ | F | n ≤· · · ≤ | F k | n k .As the Jacobson radical has the property that if a ∈ J then 1 + a is aunit, it is easy to argue that the units of R are exactly the elements thatproject to units of R/J and so | R ∗ | = | J || ( R/J ) ∗ | = | J || ( R/J ) | φ ( n , | F | ) . . . φ ( n k , | F k | ) . Finite simple rings.
We first establish some bounds in the restrictedworld of finite simple rings. Specifically, we show the following.
Proposition 2.7.
For a given threshold α > , all but finitely many finitesimple rings with Kloosterman-Salem number less than α are fields or × matrix rings over a field.Proof of Proposition 2.7. Let M be the maximal left ideal in Mat n ( F ) con-sisting of matrices with an all zero last column. Then | M | = | F | n − n and soby Theorem 2.3 we have | F | n − n ≤ C | F | n p | F | n φ R ( n, | F | ) . Hence by lemma 2.614 | F | n ( n − ≤ | F | n ( n − φ R ( n, | F | ) ≤ C . Since | F | >
1, it is clear that n is bounded. If n >
2, then | F | ≤ (4 C ) /n ( n − . Thus for any fixed C , there are finitely many choices of n , and for n > F , which proves the theorem. (cid:3) Kloosterman sums in matrix rings.
We now will eliminate the caseof Mat ( F ) in Proposition 2.7. By an explicit computation we will show thatif R = Mat ( F ), then C R ≥ | F | − | F | q (1 − | F | )(1 − | F | ) , which implies that only finitely many 2 × Proposition 2.8.
Let α ∈ (0 , ∞ ) then there are only finitely many matrixrings Mat ( F ) with Kloosterman-Salem number less than α . Thus there areonly finitely many non-field, simple rings with Kloosterman-Salem numberless than α . Let us first describe the Kloosterman sums arising from a matrix ringMat n ( F ). Recallˆ H ( A, B ) = 1 | F | n X C ∈ GL n ( F ) χ A ( − C ) χ B ( − C − )for ( A, B ) ∈ Mat n ( F ) × Mat n ( F ). We identify Mat n ( F ) × Mat n ( F ) withits Pontryagin dual in the following specific way. The trace form ( A, B ) → Tr( AB ) is F -bilinear, symmetric and non-degenerate. As Mat n ( F ) is a F -vector space, every irreducible character of Mat n ( F ) is the composition of alinear functional followed by a fixed nontrivial irreducible additive character χ of F , i.e., of the form χ ( L ( x )). As the trace form is non-degenerate, everysuch functional can be taken of the form L ( − ) = Tr( − B ) or L ( − ) = Tr( A − )for suitable A, B ∈ Mat n ( F ). Due to this we may choose an identificationof Mat n ( F ) × Mat n ( F ) with its Pontryagin dual such thatˆ H ( A, B ) = 1 | F | n X C ∈ GL n ( F ) χ ( − Tr( CA + BC − ))and we shall do so from now on.The group GL n ( F ) acts on Mat n ( F ) × Mat n ( F ) by D · ( A, B ) = (
DA, BD − )and it is easy to check that ˆ H ( A, B ) is constant on orbits. More precisely,it is a GL n ( F ) invariant: ˆ H ( DA, DB ) = ˆ H ( A, B ) . Since the trace is a similarity invariant, conjugating by C − in the definingexpression shows that ˆ H is symmetric also i.e.ˆ H ( A, B ) = ˆ H ( B, A ) . This invariance and symmetry makes the evaluation of ˆ H ( A, B ) reduce toa relatively decent number of cases based on the ranks of the matrices A and B . The non degenerate case of rank 2 matrix A reduces as ˆ H ( A, B ) =ˆ H ( I, A − B ) and the coefficientsˆ H ( I, C ) = 1 | R | X D ∈ GL ( F ) χ ( − Tr( D + CD − ))are probably the most interesting. However we will only use one particulardegenerate coefficient in our arguments:ˆ H (cid:18)(cid:20) (cid:21) , (cid:20) − (cid:21)(cid:19) = 1 | R | X D ∈ GL ( F ) χ (cid:16) − a + a ∆ (cid:17) where we write D = (cid:20) a bc d (cid:21) and ∆ = det( D ), R = Mat ( F ). To evalu-ate this we need to enumerate the distribution of upper-left entries a anddeterminants ∆ amongst the matrices in GL ( F ). ALEM SETS IN MODULES OVER FINITE RINGS 13
Partition GL ( F ) into the left cosets of SL ( F ): GL ( F ) = [ ∆ ∈ F ∗ G ∆ , where G ∆ are the matrices with determinant ∆. A simple computationshows that for ∆ = 0, there are p matrices in G ∆ with any given fixednonzero a as upper-left entry and p ( p −
1) matrices in G ∆ with upper-leftentry a = 0.For fixed ∆ ∈ F − { , } , we have X D ∈ G ∆ χ (cid:16) − a + a ∆ (cid:17) = | F | X a ∈ F ∗ χ (cid:18)(cid:18) − (cid:19) a (cid:19) + | F | ( | F | −
1) = −| F | by character orthogonality applied to χ on F . On the other hand for ∆ = 1we get X D ∈ G χ (cid:16) − a + a ∆ (cid:17) = | SL ( F ) | = ( | F | − | F | ( | F | + 1) . Putting everything together we getˆ H (cid:18)(cid:20) (cid:21) , (cid:20) − (cid:21)(cid:19) = 1 | R | X D ∈ GL ( F ) χ (cid:16) − a + a ∆ (cid:17) = 1 | R | (( | F | − | F | ( | F | + 1) − ( | F | − | F | ) . Thus | R | p | R ∗ | (cid:12)(cid:12)(cid:12)(cid:12) ˆ H (cid:18)(cid:20) (cid:21) , (cid:20) − (cid:21)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = | F | ( | F | − | F | + 1) | F | q (1 − | F | )(1 − | F | )and so the Kloosterman-Salem number of R = Mat ( F ) satisfies the claimedbound: C R ≥ | F | − | F | q (1 − | F | )(1 − | F | ) . As the right hand side goes to infinity as | F | → ∞ , we see there only finitelymany finite fields F such that the Kloosterman-Salem number of Mat ( F )lies below any given threshold. Together with Proposition 2.7, this provesProposition 2.8.2.7. The semisimple case.
We will now extend the results of the previoussections to show that all but finitely many semisimple rings with no F =Mat ( F ) factors and Kloosterman-Salem number below a given thresholdare fields. Proposition 2.9.
For any fixed < α < ∞ , all but finitely many finitesemisimple rings R with C R ≤ α and no F -factors are finite fields. Inparticular, for any fixed < α < ∞ , all but finitely many finite, odd order,semisimple rings R with C R ≤ α are finite fields. To prove Proposition 2.9, we need two lemmas. Firstly we establish auseful general lower bound on the Kloosterman-Salem number of a finitering.
Proposition 2.10.
Let R be a finite ring with = 0 and let C be itsKloosterman-Salem number. Then r < s − | R ∗ || R | ≤ C ≤ p | R ∗ | . Thus no finite ring has Kloosterman-Salem number C ≤ q .Proof. Let H be the characteristic function of the hyperbola { ( x, y ) ∈ R | xy =1 } . Plancherel’s Theorem gives | R | X m ∈ R | ˆ H ( m ) | = X x ∈ R | H ( x ) | = | H | = | R ∗ | . Thus | R | | R ∗ | | R | + X m =0 | ˆ H ( m ) | = | R ∗ | so | R ∗ || R | − | R ∗ | | R | ≤ C | R | − | R ∗ | ( | R | − ≤ C | R | − | R ∗ | and hence C ≥ s − | R ∗ || R | > s − | R | ≥ r . As | R | ˆ H ( m ) is the sum of | R ∗ | terms of modulus one, it is clear | ˆ H ( m ) | ≤| R | − | R ∗ | which yields C R ≤ p | R ∗ | . (cid:3) We will see later that the lower bound for C in Proposition 2.10 can bestrengthened to 1 while the upper bound is sharp in general.We will also need a formula for the Kloosterman-Salem number of a directproduct of rings. Proposition 2.11.
Let R = R × R be a direct product of finite rings, thentheir Kloosterman-Salem numbers are related by C R = max( C | R ∗ | , | R ∗ | C ) , ALEM SETS IN MODULES OVER FINITE RINGS 15 or equivalently C R p | R ∗ | = max C p | R ∗ | , C p | R ∗ | ! . Note that because we require our rings to have a unit, and because theKloosterman-Salem number is not defined when the dual group has no non-zero elements, this theorem only applies to non-trivial direct product de-compositions.
Proof.
First note that R ∗ = R ∗ × R ∗ . As R = R × R is also a decomposi-tion of the underlying Abelian groups, the (irreducible) characters of R areproducts of characters of R and R . The Hyperbola H ⊆ R also decom-poses as H = H × H under the decomposition R = R × R . Furthermorefor any m = ( m , m ) ∈ R × R = R it is easy to see thatˆ H ( m , m ) = 1 | R | X ( x ,x ) ∈ R ∗ × R ∗ χ m (cid:18) − (cid:18) x , x (cid:19)(cid:19) χ m (cid:18) − (cid:18) x , x (cid:19)(cid:19) = ˆ H ( m ) ˆ H ( m ) . The maximum of | ˆ H j ( m j ) | as m j varies over nonzero elements is by definition C R j | R j | − | R ∗ j | while the value of | ˆ H j (0) | is | R j | − | R ∗ j | for j = 1 ,
2. Thus it is easy tocalculate | R | p | R ∗ | max ( m ,m ) =(0 , | ˆ H ( m , m ) | = max( C R C R , C R | R ∗ | , C R | R ∗ | )as claimed by considering the three cases ( m , m ) both nonzero, m = 0and m = 0. Using the trivial bound C R j ≤ q | R ∗ j | shows that the maximumis one of the last two terms. (cid:3) Proposition 2.11 lets us construct examples to show that Z / Z factorshave a limited effect on Kloosterman-Salem numbers and explains why wehave to restrict to rings without these factors in this section. Example 2.12 (Boolean rings) . A finite Boolean ring is a direct product offinitely many Z / Z ’s. Let R n = Z / Z × · · · × Z / Z be the Boolean ring oforder 2 n , which can be identified with the ring of F -valued functions on a setof size n under the usual operations of function addition and multiplication.These rings have Kloosterman-Salem number C = 1 independent of n andhence give a sequence of rings R n with | R n | → ∞ such the Kloosterman-Salem number is uniformly bounded by 1.To prove that C = 1 for all finite Boolean rings first note that C Z / Z = 1by noting that the hyperbola consists of a single point { , } and performing a quick calculation of ˆ H ( m, n ). We then use induction and the fact that R n = Z / Z × R n − in Proposition 2.11 to find C R n = max (cid:18) × , × q | R ∗ n − | , × q | R ∗ | (cid:19) = 1 , as all Boolean rings only have one unit, the vector (1 , , . . . , J has J = 0 as | R ∗ | = | J || ( R/J ) ∗ | ≥ | J | . Thus R issemisimple. By the Chinese remainder theorem, the simple matrix factorsof R must then also have only one unit. It is easy then to see that they mustbe Mat ( F ) = Z / Z and so R ∼ = Z / Z × · · · × Z / Z is a Boolean ring. Example 2.13 (Twisting any ring by Boolean rings) . Let R be any finitering, then let S n = Z / Z × · · · × Z / Z × R be the direct product of R withthe Boolean ring of order 2 n .The product formula readily shows that C S n = max (cid:16) C R , p | R ∗ | (cid:17) = p | R ∗ | is independent of n . Thus the sequence of rings { S n } ∞ n =1 has | S n | → ∞ anduniformly bounded Kloosterman-Salem number. Despite this, these ringsexhibit the worst square root law in the sense that C S n = p | R ∗ | = p | S ∗ n | achieves the general upper bound on the Kloosterman-Salem number givenin Proposition 2.10.Now we proceed with the proof of Proposition 2.9. Proof of Proposition 2.9.
Let R be a finite semisimple ring as in equation(2.2) with no Z / Z factors.If R has only one semisimple factor, then R is simple, so by Propositions2.7 and 2.8 all but finitely many such R are fields.Now suppose that R has at least two semisimple factors, so that R =Mat n ( F ) × R , where R is a semisimple ring (note | R | ≥ = 0in our rings). Let C and C denote the Kloosterman-Salem numbers ofMat n ( F ) and R , respectively. By Proposition 2.11, we have C | R ∗ | / ≤ C R and C | GL n ( F ) | / ≤ C R . Bounding C and C below by Proposition 2.10 yields upper bounds for | GL n ( F ) | and | R ∗ | : | GL n ( F ) | , | R ∗ | ≤ C R . If R is a product of k matrix rings, as in (2.2), the previous equation impliesthat | GL n ( F ) | · · · | GL n k ( F k ) | ≤ C R . This implies that 14 | F j | n j ≤ | GL n j ( F j ) | ≤ C R ALEM SETS IN MODULES OVER FINITE RINGS 17 for j = 1 , . . . , k , so n j and | F j | are bounded for all j . Further, as F j = Z / Z ,we have | GL n j ( F j ) | ≥
2, hence2 k − ≤ | GL n ( F ) | · · · | GL n k ( F k ) | ≤ C R , which shows that k is bounded. As the size and number of R ’s semisimplefactors are bounded in terms of C R , it follows that | R | is bounded in termsof C R .It follows that there are finitely many semisimple R with no Z / Z factorsand with more than one semisimple factor and Kloosterman-Salem number C R ≤ α , which concludes the proof. (cid:3) Remark . The same proof shows that all but finitely many semi-simplerings R with C R ≤ α and a bounded number F factors (say ≤ n such factors)are fields.2.8. Finite rings with Jacobson radical.
In this section we show that atmost finitely many (odd) rings with Kloosterman-Salem number less than α >
Lemma 2.15.
Let R be a finite ring with Jacobson radical J , and let S = R/J be the semisimple part of R , so that we have a short exact sequence ofrings and ideals → J → R → S → . If C R and C S = C R/J denote the Kloosterman-Salem numbers of R and S ,then C R ≥ C S | J | / = C R/J | J | / . Proof.
Let χ m , χ n be any additive characters of S (at least one of themnontrivial), pulling them back under the quotient map π : R → S , one canview them as additive characters of R which are equal to 1 on J . Usingthese characters one obtains certain Kloosterman sumsˆ H ( m, n ) = 1 | R | X x ∈ R ∗ χ m ( − x ) χ n ( − x ) , which represent certain Fourier coefficients for the hyperbola of R . As χ m ( − x ) and χ n ( − x ) only depend on the image of x, x in S , using that R ∗ = π − ( S ∗ ), this sum degenerates intoˆ H ( m, n ) = | J || R | X x ∈ S ∗ χ m ( − x ) χ n ( − x ) . Taking the maximum over ( m, n ) = (0 , ∈ S × S one getsmax ( m,n ) ∈ S × S | ˆ H ( m, n ) | = | J || R | C S | S ∗ | and so C R | R | − | R ∗ | ≥ | J || R | C S | S ∗ | . (Note this last inequality is not necessarily an equality as the previous max-imum was only over characters of R induced from S and not all charactersof R .) Using that | R ∗ | = | S ∗ || J | this simplifies to give C R ≥ C S | J | / = C R/J | J | / , as was to be shown. (cid:3) We are now ready to show that there are finitely many finite rings withno Z / Z semisimple factors that have a non-zero Jacobson radical and aKloosterman-Salem number below a given threshold. This is the last stepin the proof of the main theorem. Proof of J = 0 case. Let R be a finite ring with Kloosterman-Salem number C R bounded above by some threshold α ∈ (0 , ∞ ); further, assume that R has no Z / Z semisimple facters.It follows from Lemma 2.15 and Proposition 3.2 that | J | ≤ α and C R/J ≤ α . As R/J is semisimple and C R/J is bounded by α , Proposition 2.9 impliesthat R/J must be a field in all but finitely many cases. Thus we may assumethat we have a short exact sequence0 → J → R → F → , where F is a finite field. It remains to be shown that the Jacobson radicalis zero in all but finitely many cases.If J = 0, then J/J = 0 by Nakayama’s lemma. As J/J is a non-zero R/J = F -vector space, it follows that | F | divides | J | . Since | J | is boundedby α , it follows that | F | is also bounded by α , hence | R | is bounded by α ,which implies that only a finite number of rings satisfy these conditions. (cid:3) Thus we have established Theorem and in view of the above, Theorem1.2 follows. 3.
Quantitative Results
The proofs of Propositions 2.7 and 2.8 yield explicit bounds which werecord in the following corollary.
Corollary 3.1.
Let F be a finite field, let R be the finite simple ring Mat n ( F ) , and let C R be the Kloosterman-Salem number of R .If n = 2 , (3.1) C R ≥ | F | − | F | . If n ≥ , (3.2) C R ≥ | F | n ( n − . ALEM SETS IN MODULES OVER FINITE RINGS 19
In particular, C R ≥ √ for all F and all n . In addition, we have (3.3) n ≤ p ( C R ) + 2 . Proof. If n = 2, then by Proposition 2.8, C R ≥ | F | − | F | p φ (2 , F ) ≥ | F | − | F | , which proves (3.1).Now suppose that n ≥
3. In this case the proof of Proposition 2.7 showsthat(3.4) 14 | F | n ( n − ≤ | F | n ( n − φ ( n, | F | ) ≤ C R . Taking square roots proves (3.2).To prove the bound (3.3) on n , we combine trivial bound 2 ≤ | F | with(3.4): 2 n ( n − − ≤ C . For n ≥
3, we have ( n − ≤ n ( n − −
2, so2 ( n − ≤ C , which implies (3.3) by taking logarithms. For n = 2, the upper bound istrivial and so the proof is complete. (cid:3) Using Lemma 2.15 we can strengthen the general lower bound for Kloosterman-Salem numbers obtained in Proposition 2.10.
Proposition 3.2.
Let R be a finite ring with Kloosterman-Salem number C R . Then C R ≥ with equality if and only if R is a finite Boolean ring.Any non-Boolean finite ring has C R ≥ √ .Proof. Since C R ≥ C R/J | J | , Lemma 2.15 implies that it is enough to con-sider the semisimple case J = 0. The product formula in Proposition 2.11reduces to the case where R = Mat n ( F ). By Corollary 3.1, if n ≥ C R ≥ √ R = Mat ( F ) = F a finite field. Hereby a result on Kloosterman sums, which we cite below, we have C F ≥ √ F . A simple direct computation shows C F = 1as explained in Example 2.12.Furthermore it is easy to see from these arguments that C R = 1 if and onlyif R is semisimple with all simple factors F , a Boolean ring, and otherwise C R ≥ √ (cid:3) The lower bound C F ≥ √
2, where F is a finite field, is implicit in theoriginal work of Kloosterman [9]. A modern proof can be found on page 22of [10], where it is shown that if | F | = q , then(3.5) C F ≥ q − q − q − q − q − q − . For q >
3, the right hand side of (3.5) is greater than 2, and one may easilycheck that C F = √
2. 4.
Extremal Rings
In this section we study rings that have the worst possible square rootlaw for Kloosterman sums. We call such rings extremal:
Definition 4.1.
A finite ring R is extremal if its Kloosterman-Salem number C R achieves the general upper bound C R = p | R ∗ | .Example 2.12 shows that Boolean rings F N are extremal, and exam-ple 2.13 shows that we can create extremal rings by taking products withBoolean rings. It turns out that there are further examples, which we willpartially classify.We begin by providing an alternate characterization of extremal rings. Theorem 4.2.
Let R be a finite ring with Kloosterman-Salem number C .Then the following are equivalent:(1) C < p | R ∗ | .(2) R is not an extremal ring.(3) The subset { ( x − , x − −
1) : x ∈ R ∗ } of R generates R as an additivegroup.(4) For every A, B ∈ R , there exists a positive integer n and units x , . . . , x n ∈ R ∗ such that x + · · · + x n − n = Ax − + · · · + x − n − n = B Proof.
Note that (1) and (2) are equivalent by definition, and (3) and (4)are equivalent as R × R is finite and so a subset S ⊆ R generates it as anadditive group if and only if it generates it as a semigroup. In this case thismeans any ( A, B ) ∈ R is a finite sum of elements of the form ( x i , x − i ).Thus it remains to prove the equivalence of (2) and (3).Suppose R is an extremal ring, which means that C R = p | R | ∗ . Thishappens if and only if there exists ( m, n ) = (0 ,
0) such that | K ( m, n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ R ∗ χ m ( x ) χ n ( x − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | R ∗ | . That is, K ( m, n ) has no cancellation. As | χ m ( x ) χ n ( x − ) | = 1, this can onlyhappen if χ m ( x ) χ n ( x − ) is constant for x ∈ R ∗ .Since χ = χ m ⊗ χ n : R × R → C is a non-trivial additive character,its kernel K is a proper subgroup of R × R . Elements in R × R havethe same χ -value if and only if they lie in the same coset of K , and so ALEM SETS IN MODULES OVER FINITE RINGS 21 the Kloosterman sum K ( m, n ) has no cancellation only if the hyperbola H = { ( x, x − ) | x ∈ R ∗ } lies in a single coset of K .Since H − (1 ,
1) is contained in K − (1 , H − (1 ,
1) cannotgenerate R under addition. Conversely, if H − (1 ,
1) generates a propersubgroup K of R then a pullback character under π : R × R → ( R × R ) /K yields a non-trivial additive character χ m ⊗ χ n of R × R for which K ( m, n )has no cancellation. (cid:3) Corollary 4.3. If R is a finite ring with Jacobson radical J then if thesemisimple ring R/J is extremal, this implies R itself is extremal.Proof. First recall the hyperbola H in R × R maps onto the hyperbola ¯ H in R/J × R/J under the quotient map. Thus if H generates R × R as anadditive group, ¯ H will generate R/J × R/J as an additive group. Thusby Theorem 4.2, R not extremal implies R/J is not extremal. The resultfollows by taking the contrapositive. (cid:3)
Corollary 4.3 reduces questions about extremal rings to questions aboutextremal semisimple rings.
Corollary 4.4. If R = R × · · · × R n is a direct product of finite rings then R is extremal if and only if at least one of the R i , ≤ i ≤ n is extremal.Proof. One can prove this either using the product formula for Kloosterman-Salem numbers or by noting that the hyperbola H in R × R is the directproduct of the hyperbolas H i in R i × R i . Thus H − { (1 , } generates R × R as an additive group if and only if each H i − { (1 i , i ) } generates R i × R i asan additive group. Thus R is not extremal if and only if all the R i ’s are notextremal. (cid:3) Corollary 4.4 lets us reduce the questions about extremal semisimple ringsto ones about extremal simple rings, i.e., Mat n ( F ) where F is a finite field.We deal with fields next. Proposition 4.5.
Let F q be the finite field of order q . Then F q is extremalif and only if q = 2 , , .Proof. Let C be the Koosterman-Salem number of F q . By the Weil bound(2.1), any nontrivial Kloosterman sum is bounded by 2 √ q . Thus the fieldis not extremal as long as 2 √ q < q − q − q − q + 1 > q >
5. Thus any finite field of size q > F has C = 1 and only one unit so it is extremal. The field F has C = √ p | F ∗ | so it is extremal.For F the Kloosterman sums are given by K ( m, n ) = P x ∈ F ∗ χ ( mx + nx ).The x = 1 , − x = 2 , − K ( m, n ) = 2 cos(2 π ( m + n ) /
5) + 2 cos(4 π ( m − n ) / . As it is impossible to have m + n = 0 = m − n without m = n = 0 in F we see that F is not extremal.Finally write F = F [ u ] where u is a primitive third root of unity andhence solves u + u + 1 = 0. Recall the trace Tr : F → F is given byTr( a + bu ) = ( a + bu ) + ( a + bu ) = 2 a + b ( u + u ) = b for any a, b ∈ F ,as the Galois group of F over F is cyclic of order two generated by theFrobenius map Frob : x → x . The Kloosterman sum is then given by K ( m, n ) = X x ∈ F ∗ χ (cid:16) Tr (cid:16) mx + nx (cid:17)(cid:17) where χ ( s ) = e πix is the nontrivial additive character of F . Thus K ( m, n ) = χ (Tr( m + n )) + χ (Tr( mu + n (1 + u ))) + χ (Tr( m (1 + u ) + nu )) . It follows that K (1 ,
1) = χ (0) + χ (0) + χ (0) = 3 = | F ∗ | and so F isextremal. (cid:3) Hyperbola Graphs
Let R be a finite ring and let S be a subset of R d for some d ≥
1. We say S is symmetric if x ∈ S implies − x ∈ S . Please consult [13] for the graphtheoretic background needed in this section. Definition 5.1.
Given a symmetric set S ⊆ R d for some d ≥
1. We definethe S -graph G S to be the graph whose vertex set is V = R d and where v and v are joined by a single edge in S if and only if v − v ∈ S .Note this graph has no multiple edges, and has loops if and only if 0 ∈ S .It is a regular graph where each vertex has degree d = | S | . Graphs of thesesort have been studied extensively [3, 2].Recall the adjacency matrix A of this graph is a | V | × | V | matrix whoserows and columns are indexed by the vertices of the graph and where a ij = 1if vertex v i is joined to vertex v j by an edge and a ij = 0 if not.We first relate the spectrum of the graph G S , the set of eigenvalues of A ,to the Fourier coefficients of the characteristic function of the set S . Proposition 5.2.
Let G S be the S -graph of a symmetric set S ⊆ R d andlet A be its adjacency matrix. The eigenvectors of A are exactly the charac-ters of the additive group of the ring R and the character χ m corresponds toeigenvalue | R | d ˆ S ( m ) , where ˆ S ( m ) is the Fourier coefficient of the character-istic function S with respect to that character. Thus the spectrum of A is thesame as the set of Fourier coefficients of S scaled by | R d | . In particular thespectral gap between the largest eigenvalue and one of 2nd largest magnitudeis | S | − max m =0 | R d || ˆ S ( m ) | . Proof.
First note that we may think of a function f : V = R d → C asa column vector whose entries are indexed by the vertex set V = R d and ALEM SETS IN MODULES OVER FINITE RINGS 23 whose v -th entry is f ( v ). Under this identification, it is easy to check thatthe adjacency matrix A corresponds to an operator g = Af where g ( v ) = X u ∈ S f ( v + u ) . Now let f = χ m be an additive character of R d , then Af ( v ) = X u ∈ S χ m ( v + u )= X u ∈ R d χ m ( v ) χ m ( u ) S ( u )= χ m ( v ) X u ∈ R d χ m ( − u ) S ( u )= | R d | ˆ S ( m ) χ m ( v )for all v ∈ V . Thus Af = | R | d ˆ S ( m ) f and f = χ m is an eigenvector of A with eigenvalue | R | d ˆ S ( m ). As ( R d , +) is a finite abelian group, the numberof such characters is | R d | = | V | . As irreducible characters of finite groups arelinearly independent, we see that we have indeed found all the eigenvectorsof A . The proposition follows. (cid:3) In a regular graph of degree d , d is the largest eigenvalue of A . It is alsoan eigenvalue of maximal magnitude though − d is also in the spectrum andof equal magnitude if the graph is bipartite. Furthermore by a theorem ofFrobenius, the multiplicity of d as an eigenvalue of A is the same as thenumber of connected components of the graph. Thus we have the followingcorollary: Corollary 5.3.
Let G S be the S -graph arising from a symmetric set S ⊆ R d .Then G S is connected if and only if max m =0 ˆ S ( m ) < | R | − d | S | . Furthermore we have max m =0 | ˆ S ( m ) | < | R | − d | S | if and only if the graph is connected and not bipartite.Proof. The first part follows from the Theorem of Frobenius mentioned inthe preceding paragraph. The second part then follows as the only elementof the spectrum that can have the same magnitude as d besides d itself is − d and − d is in the spectrum of a connected regular graph if and only ifthe graph is bipartite. (cid:3) Corollary 5.4. If R is a finite ring which is not extremal then for any A, B ∈ R there exists n ≥ and units u , . . . , u n such that A = u + · · · + u n B = u − + · · · + u − n . Proof. If R is not extremal, by Corollary 5.3 we have that the hyperbolagraph is a connected graph. Thus in particular it is possible to get fromvertex (0 ,
0) to vertex (
A, B ) with a simple path. This means that (
A, B ) =(0 ,
0) + ( u , u − ) + · · · + ( u n , u − n ) for some ( u j , u − j ) on the hyperbola. Thisgives the result. (cid:3) Definition 5.5.
Let R be a finite ring and let H ⊆ R be the hyperbola H = { ( u, u − ) | u ∈ R ∗ } . The hyperbola graph is the graph arising from thesymmetric set H . By the earlier results of this section, this graph is regularof degree d = | R ∗ | and has a spectrum given by | R | times the Fouriercoefficients of H . Corollary 5.6.
Let R be a finite ring and C be its Kloosterman-Salem num-ber. Then if R is not extremal, the hyperbola graph G H is connected andnot bipartite. Furthermore the spectral gap is given by | R ∗ | − p | R ∗ | C . Con-versely when R is extremal, the hyperbola graph G H is either disconnectedor connected and bipartite.Proof. The spectrum of a regular graph of degree d is real and contained inthe interval [ − d, d ]. It is connected if and only if d has multiplicity 1 as aneigenvalue and bipartite if and only if − d is an eigenvalue.When R is not extremal, | R || ˆ H ( m ) | < | R ∗ | for m = 0 and so d = | R ∗ | hasmultiplicity one as an eigenvalue and − d does not occur as an eigenvalue.Furthermore by definition | R | max m =0 | ˆ H ( m ) | = C p | R ∗ | , so the spectral gap of the hyperbola graph is given by | R ∗ | − C p | R ∗ | and the corollary follows. (cid:3) Example 5.7 (Hyperbola graphs of extremal examples) . Let K n denote thecomplete graph on n vertices. The hyperbola graphs of the extremal rings F , F , and F are disjoint unions of complete graphs: • The hyperbola graph of F is the disjoint union of two edges, that is,two K graphs. • The hyperbola graph of F is the disjoint union of 3 triangles, that is,three K graphs. • The hyperbola graph of F is the disjoint union of four K ’s. ALEM SETS IN MODULES OVER FINITE RINGS 25
For q > F q is not an extremal ring and so the associated hyperbolagraphs are connected, non-bipartite graphs; thus the pattern exhibited by F , F and F does not continue.Explicitly for F , the hyperbola is given by H = { (1 , , (2 , , (3 , , (4 , } .Thus given ( x, y ) ∈ F it is clear there is a path from ( x, y ) to all the( x + n, y + n ) , n = 0 , , , , , ∈ H repeatedly to( x, y ). On the other hand, adding (2 ,
3) or (3 ,
2) to ( x, y ) raises or lowersthe value of y − x by one. From these facts it is easy to directly check thatthe hyperbola graph of F is connected. In fact the 5 vertices on the line y − x = b for fixed b form a cycle subgraph C . The hyperbola graph of F is obtained from the five cycle subgraphs for b = 0 , , , , y − x = b to exactlyone point in the cycle subgraph corresponding to the line y − x = b + 1 andto exactly one point in the cycle subgraph for the line y − x = b − Example 5.8. If R , R are finite rings and R = R × R is their directproduct, the Chinese remainder theorem shows that H = H × H where H is the hyperbola of R and H j is the hyperbola of R j . The resulting hyperbolagraph G H has ( x , y ) adjacent to ( x , y ) if and only if x , x are adjacentin G H and y , y are adjacent in G H . Thus the adjacency matrix of G H is the tensor product of those for G H and G H . Thus if λ , . . . , λ N is thespectrum of G H (listed with multiplicity) and µ , . . . , µ K is the spectrumof G H then λ i µ j , ≤ i ≤ N, ≤ j ≤ K is the spectrum of G H .Given a graph, a random walk on the graph is a process where we start atsome vertex and at each step move to an adjacent vertex in a manner whereit is equally likely that we move to any adjacent vertex versus any other.It is well known (see [13]) that the random walk on a connected, nonbipartite, regular graph converges to the uniform distribution. This meansthat no matter where we start, after a large number of random steps, we areequally likely to be anywhere in the graph. More precisely, in the hyperbolagraph for a non-extremal ring R , using the results in [13], we have if p tij isthe probability that starting at vertex i we end up at vertex j after t stepsin a random walk, then p tij satisfies (cid:12)(cid:12)(cid:12)(cid:12) p tij − | R | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C p | R ∗ | ! t . where C is the Kloosterman-Salem number of the finite ring R .A d -regular, connected, non-bipartite graph has good expansion proper-ties if its spectral gap is large. In particular, if d − λ ≥ ǫd then G is an ǫ -expander (see [13]). It follows that: Corollary 5.9. If R is a non-extremal ring with Kloosterman Salem number C , then the corresponding hyperbola graph is a | R ∗ | -regular, connected, non-bipartite simple graph and is an expander graph with expander ratio is ǫ = 12 − C p | R ∗ | ! . Remark . Among expander graphs, the Ramanujan graphs are thosewith best spectral expansion. The hyperbola graph of a non-extremal ring R is a Ramanujan graph if λ ≤ p | R ∗ | −
1. This happens if and only ifthe Kloosterman-Salem number C satisfies C ≤ q − | R ∗ | . If R is an oddorder ring, our results show that aside from a finite set of exceptions, thiscan only occur when R is a field.Further, a graph is Ramanujan if and only if its Ihara zeta function sat-isfied the “Riemann Hypothesis” [15]. The Ihara zeta function is definedfor all graphs, and so it provides a zeta function associated to Kloostermansums over general rings. For Kloosterman sums over fields, the Ihara zetafunction and the classical zeta function ([7] section 11.5) are closely related.The results of section 2.2 yield an upper bound on the independencenumber of hyperbola graphs.
Proposition 5.11.
The independence number of the hyperbola graph of afinite ring R with Kloosterman-Salem number C R is at most C R | R | √ | R ∗ | .Proof. Let E ⊂ R be an independent set. This means that there are nosolutions to x − y ∈ H with x and y in E . In the language of section 2.2,this means that n ( E ) = 0, hence by Theorem 2.2 | E | ≤ C R | R | p | R ∗ | . (cid:3) This bound implies a lower bound on the chromatic number of hyperbolagraphs.
Proposition 5.12.
The chromatic number of the hyperbola graph of a finitering R with Kloosterman-Salem number C R is at least √ | R ∗ | C R .Proof. Suppose that the hyperbola graph can be colored by k colors so thatno two adjacent vertices are the same color. This partitions the vertex set R into k sets E , . . . , E k , where each E i is monochromatic. Since verticesof the same color are not connected, each E i is an independent set, and soby Proposition 5.11, we have | R | = k X i =1 | E i | ≤ k C R | R | p | R ∗ | . ALEM SETS IN MODULES OVER FINITE RINGS 27
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A. Iosevich, Department of Mathematics, University of Rochester, Rochester,NY 14627
E-mail address : [email protected] B. Murphy, Department of Mathematics, University of Rochester, Rochester,NY 14627
E-mail address : [email protected]@math.rochester.edu