aa r X i v : . [ m a t h . F A ] M a y A zero- p law for cosine families Jean Esterle
Abstract : Let a ∈ R , and let k ( a ) be the largest constant such that sup | cos ( na ) − cos ( nb ) | < k ( a ) for b ∈ R implies that b ∈ ± a + π Z . We show that if a cosinesequence ( C ( n )) n ∈ Z with values in a Banach algebra A satisfies sup n ≥ k C ( n ) − cos ( na ).1 A k < k ( a ), then C ( n ) = cos ( na ) for n ∈ Z . Since p ≤ k ( a ) ≤ p forevery a ∈ R , this shows that if some cosine family ( C ( g )) g ∈ G over an abeliangroup G in a Banach algebra satisfies sup g ∈ G k C ( g ) − c ( g ) k < p for some sca-lar cosine family ( c ( g )) g ∈ G , then C ( g ) = c ( g ) for g ∈ G , and the constant p isoptimal. We also describe the set of all real numbers a ∈ [0, π ] satisfying k ( a ) ≤ . Keywords : Cosine function, scalar cosine function, commutative local Ba-nach algebra, Kronecker’s theorem, cyclotomic polynomialsAMS classification : Primary 46J45, 47D09, Secondary 26A99
Let G be an abelian group. Recall that a G -cosine family of elements of aunital normed algebra A with unit element 1 A is a family ( C ( g )) g ∈ G of elementsof A satisfying the so-called d’Alembert equation C = A , C ( g + h ) + C ( g − h ) = C ( g ) C ( h ) ( g ∈ G , h ∈ G ). (1)A R -cosine family is called a cosine function, and a Z -cosine family is calleda cosine sequence.A cosine family C = ( C ( g )) g ∈ G is said to be bounded if there exists M > k C ( g ) k ≤ M for every g ∈ G . In this case we set k C k ∞ = sup g ∈ G k C ( g ) k , d i st ( C , C ) = k C − C k ∞ .A cosine family is said to be scalar if C ( g ) ∈ C .1 A for every g ∈ G . It is easy to seeand well-known that a bounded scalar cosine sequence satisfies C ( n ) = cos ( an )for some a ∈ R . 1trongly continuous operator valued cosine functions are a classical tool inthe study of differential equations, see for example [2], [3], [15], [19], and a func-tional calculus approach to these objects was developped recently in [11].Bobrowski and Chojnacki proved in [4] that if a strongly continuous opera-tor valued cosine function on a Banach space ( C ( t )) t ∈ R satisfies sup t ≥ k C ( t ) − c ( t ) k < c ( t ) then C ( t ) = c ( t ) pour t ∈ R , and Zwart and F. Schwenninger showed in [18] that this resultremains valid under the condition sup t ≥ k C ( t ) − c ( t ) k <
1. The proofs were ba-sed on rather involved arguments from operator theory and semigroup theory.Very recently, Bobrowski, Chojnacki and Gregosiewicz [5] showed more preci-sely that if a cosine function C = C ( t ) satisfies sup t ∈ R k C ( t ) − c ( t ) k < p for somescalar bounded continuous cosine function c ( t ), then C ( t ) = c ( t ) for t ∈ R , wi-thout any continuity assumption on C , and the same result was obtained inde-pendently by the author in [10]. The constant p is obviously optimal, sincesup t ∈ R | cos ( at ) − cos (3 at ) | = p for every a ∈ R \ {0}.The author also proved in [10] that if a cosine sequence ( C ( t )) t ∈ R satisfiessup t ∈ R k C ( t ) − cos ( at )1 A k = m < a
0, then the closed algebra ge-nerated by ( C ( t )) t ∈ R is isomorphic to C k for some k ≥
1, and that there existsa finite family p , . . . , p k of pairwise orthogonal idempotents of A and a family( b , . . . , b k ) of distinct elements of the finite set ∆ ( a , m ) : = { b ≥ sup t ∈ R | cos ( bt ) − cos ( at ) | ≤ m } such that we have C ( t ) = k X j = cos ( b j t ) p j ( j ∈ R ).Also Chojnacki developped in [7] an elementary argument to show that if( C ( n )) n ∈ Z is a cosine sequence in a unital normed algebra A satisfying sup n ≥ k C ( n ) − c ( n ) k < c ( n )) n ∈ Z then c ( n ) = C ( n ) for every n ,which obviously implies the result of Zwart and F. Schwenninger. His approach isbased on an elaborated adaptation of a very short elementary argument used byWallen in [20] to prove an improvement of the classical Cox-Nakamura-Yoshida-Hirschfeld-Wallen theorem [8], [13], [16] which shows that if an element a of aunital normed algebra A satisfies sup n ≥ k a n − k <
1, then a = C ( ng ) and c ( ng ) for g ∈ G , Cho-najcki observed in [7] that if a cosine family C ( g ) satisfies sup g ∈ G k C ( g ) − c ( g ) k < c ( g ) then C ( g ) = c ( g ) for every g ∈ G .In the same direction Schwenninger and Zwart showed in [17] that if a cosinesequence ( C ( n )) n ∈ Z in a Banach algebra A satisfies sup n ≥ k C ( n ) − A k < , then C ( n ) = A for every n .The purpose of this paper is to obtain optimal results of this type. We prove2 "zero- p " law : if a cosine family ( C ( g )) g ∈ G satisfies sup g ∈ G k C ( g ) − c ( g ) k < p for some scalar cosine family ( c ( g )) g ∈ G then C ( g ) = c ( g ) for every g ∈ G . Sincesup n ≥ ¯¯ cos ¡ n π ¢ − cos ¡ n π ¢¯¯ = cos ¡ π ¢ − cos ¡ π ¢ = p , the constant p is opti-mal.In fact for every a ∈ R there exists a largest constant k ( a ) such that sup n ≥ | cos ( nb ) − cos ( na ) | < k ( a ) implies that cos ( nb ) = cos ( na ) for n ≥
1, and we prove thatif a cosine sequence ( C ( n )) n ∈ Z in a Banach algebra A satisfies sup n ≥ | C ( n ) − cos ( na )1 A | < k ( a ) then C ( n ) = cos ( na ) for n ≥
1. This follows from the followingresult, proved by the author in [10].
Theorem 1.1.
Let ( C ( n )) n ∈ Z be a bounded cosine sequence in a Banach algebra. Ifspec ( C (1)) is a singleton, then the sequence ( C ( n )) n ∈ Z is scalar, and so there existsa ∈ R such that C ( n ) = cos ( na ) for n ≥ k ( a ). As mentioned above, it follows from [17] that k (0) = , and it isobvious that k ( a ) ≤ sup n ≥ | cos ( na ) − cos (3 na ) | ≤ p if a ∉ π Z . We observe that k ( a ) = p if a π is irrational, and we prove, using basic results about cyclotomicfields, that k ( a ) < p if a π is rational.We also show that the set Ω ( m ) : = { a ∈ [0, π ] | k ( a ) ≤ m } is finite for every m < p . We describe in detail the set Ω ¡ ¢ : it contains 43 elements, and theonly values for k ( a ) for which k ( a ) < are p = cos ¡ π ¢ + cos ¡ π ¢ ≈ p = cos ¡ π ¢ + cos ¡ π ¢ ≈ cos ¡ π ¢ + cos ¡ π ¢ ≈ p law follows then from the fact that k ( a ) ≥ cos ¡ π ¢ + cos ¡ π ¢ = p for every a ∈ R .We also show that given a ∈ R and m < Γ ( a , m ) of scalar cosinesequences ( c ( n )) n ∈ Z satisfying sup n ∈ Z | c ( n ) − cos ( na ) | ≤ m is finite. This impliesthat if a cosine sequence ( C ( n )) n ∈ Z satisfies sup n ∈ Z k C ( n ) − cos ( an )1 A k ≤ m , thenthere exists k ≤ c ar d ( Γ ( a , m )) such that the closed algebra generated by ( C ( n )) n ∈ Z is isomorphic to C k , and there exists a finite family p , . . . , p k of pairwise or-thogonal idempotents of A and a finite family c , . . . , c k of distinct elements of Γ ( a , m ) such that we have C ( n ) = k X j = c j ( n ) p j ( n ∈ Z ).This last result does not extend to cosine families over general abelian group.Let G = ( Z /3 Z ) N : we give an easy example of a G -cosine family ( C ( g )) g ∈ G withvalues in l ∞ such that the closed subalgebra generated by ( C ( g )) g ∈ G equals l ∞ ,while sup g ∈ G k l ∞ − C ( g ) k = . 3he author warmly thanks Christine Bachoc and Pierre Parent for giving himthe arguments from number theory which lead to a simple proof of the fact that k ( a ) < p if a ∉ π Q . We introduce the following notation, to be used throughout the paper.
Definition 2.1.
Let a ∈ π Q . The order of a, denoted by or d ( a ), is the smallestinteger u ≥ such that e iua = S of the unit circle T is said to be independent if z n . . . z n k k z , . . . , z k ) of distincts elements of S and every family( n , . . . , n k ) ∈ Z k of such that z j j ≤ j ≤ k . It follows from a classical theo-rem of Kronecker, see for example [14], page 21 that if S = { z , . . . , z k } is a finiteindependent set then the sequence ( z n , . . . , z nk ) n ≥ is dense in T k . We deducefrom Kronecker’s theorem the following observation. Proposition 2.2.
Let a ∈ [0, π ]. For m ≥ set Γ ( a , m ) = © b ∈ [0, π ] : sup n ≥ | cos ( na ) − cos ( nb ) | ≤ m ª . Then Γ ( a , m ) is finite for every m < m ∈ [1, 2). Notice that if b ∈ R , and if the set © e i a , e ib ª is inde-pendent, then it follows from Kronecker’s theorem that the sequence ¡¡ e ina , e inb ¢¢ n ≥ is dense in T , and so sup n ≥ | cos ( na ) − cos ( nb ) | =
2, and b ∉ Γ ( a , m ).Suppose that a π ∈ Q , and denote by u the order of a , so that e iua =
1. If b π ∉ Q ,then the sequence ¡ e iunb ¢ n ≥ is dense in T , and so2 ≥ sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ | − cos ( nub ) | = b ∉ Γ ( a , m ).The same argument shows that if a π ∉ Q , and if b π ∈ Q , then b ∉ Γ ( a , m ). So weare left with two situations1) a π ∉ Q , and there exists p q k ∈ Z such that bq = ap + k π .2) a π ∈ Q and b π ∈ Q .We consider the first case. Replacing b ∈ [0, π ] by − b ∈ [ − π , 0] if necessary wecan assume that p ≥ q ≥
1, and we can assume that we have qb = p a + k π r ,4ith g cd ( p , q ) = r ≥ g cd ( r , k ) = k ra π ∉ Q , sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ ¯¯ cos ( nr q a ) − cos ( nr qb ) ¯¯ = sup n ≥ ¯¯ cos ( nr q a ) − cos ( nr p a ) ¯¯ = sup t ∈ R ¯¯ cos ( q t ) − cos ( p t ) ¯¯ ,Since g cd ( p , q ) =
1, we have sup t ∈ R ¯¯ cos ( q t ) − cos ( p t ) ¯¯ = p or q is even,so we can assume that p and q are odd. Set s = q − .It follows from Bezout’s identity that there exist n ≥ e inp π q = e is π q and setting t = n π q , we obtain sup t ∈ R ¯¯ cos ( q t ) − cos ( p t ) ¯¯ ≥ − cos µ s π s + ¶ = + cos µ π q ¶ .The same argument shows that we have sup t ∈ R ¯¯ cos ( q t ) − cos ( p t ) ¯¯ ≥ + cos µ π p ¶ .We obtain p ≤ π ar ccos ( m − q ≤ π ar ccos ( m − sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ ¯¯ cos ( nq a ) − cos ( nqb ) ¯¯ = sup n ≥ ¯¯¯¯ cos ( nq a ) − cos µ np a + nk q π r ¶¯¯¯¯ .Assume that k
0, set d = g cd ( r , q ), r = rd , q = qd . Then g cd ( k q , r ) = u ≥ ukq π r ∈ π r + π Z . This gives sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ ¯¯¯¯ cos ( nuq a ) − cos µ npua + n π r ¶¯¯¯¯ .If r is even, set r = r . We obtain sup n ≥ ¯¯¯¯ cos ( nuq a ) − cos µ npua + n π r ¶¯¯¯¯ ≥ sup n ≥ ¯¯ cos ((2 n + r uq a ) − cos ¡ (2 n + r up a ) + π ¢¯¯ .5ince 2 r ua ∉ π Q , there exists a sequence ( n j ) j ≥ of integers such that l i m j →+∞ ¯¯¯ e i n j r ua + ir ua ¯¯¯ = l i m j →+∞ ¯¯ cos ((2 n j + r uq a ) − cos ¡ (2 n j + r up a ) + π ¢¯¯ = sup n ≥ | cos ( na ) − cos ( nb ) | = r is odd. Set r = r − . The same calculation as abovegives sup n ≥ ¯¯¯¯ cos ( nuq a ) − cos µ npua + n π r ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos (( n (2 r + + r ) uq a ) − cos µ ( n (2 r + + r ) up a + n (2 r + + r )2 r + π ¶¯¯¯¯ ≥ + cos µ π r + ¶ .Hence r = r + ≤ π arccos ( m − , r = r d ≤ r q ≤ ³ π arccos ( m − ´ .This gives2 | k | π ≤ r | qb − p a | ≤ π µ π ar ccos ( m − ¶ , | k | ≤ µ π ar ccos ( m − ¶ .We see that Γ ( a , m ) is finite if a π ∉ Q , and that we have c ar d ( Γ ( a , m )) ≤ µ π ar ccos ( m − ¶ .Now consider the case where a π ∈ Q , b π ∈ Q . We first discuss the case where a = b
0. We have b = p π q , where 1 ≤ p ≤ q , g cd ( p , q ) = p = q =
1, then b = π , and sup n ≥ | − cos ( n π ) | =
2. So we may assume that p ≤ q −
1. If p is odd, then we have sup n ≥ | − cos ( nb ) | ≥ | − cos ( qb ) | = − cos ( p π ) = p is even, so that q is odd. Set r = q − . There exists n ≥ r ∈ Z such that n p − r ∈ q Z , and we have sup n ∈ Z | − cos ( nb ) | ≥ | − cos (2 n b ) | = ¯¯¯¯ − cos µ r π r + ¶¯¯¯¯ = + cos µ π q ¶ .We obtain again q ≤ π arccos ( m − , and c ar d ( Γ (0, m )) ≤ ³ π arccos ( m − ´ .6ow assume that a
0, and let u ≥ a . We have sup n ≥ | − cos ( nub ) | = sup n ≥ | cos ( nua ) − cos ( nub ) | ≤ m ,and so there exists there exists c ∈ Γ (0, m ) such that cos ( nc ) = cos ( nub ) for n ≥ cos ( c ) = cos ( ub ), and b = ± cu + k π u , where k ∈ Z . We obtain c ar d ( Γ ( a , m )) ≤ uc ar d ( Γ (0, m )) ≤ uc ar d µ π ar ccos ( m − ¶ . ä We do not know whether it is possible to obtain a majorant for c ar d ( Γ ( a , m ))which depends only on m when a ∈ π Q . Theorem 2.3.
Let a ∈ R , let m < and let ( C ( n )) n ∈ Z be a cosine sequence ina Banach algebra A such that sup n ≥ k C ( n ) − cos ( na ) k ≤ m . Then there existsk ≤ c ar d ( Γ ( a , m )) such that the closed algebra generated by ( C ( n )) n ∈ Z is isomor-phic to C k , and there exists a finite family p , . . . , p k of pairwise orthogonal idem-potents of A and a finite family b , . . . , b k of distinct elements of Γ ( a , m ) such thatwe have C ( n ) = k X j = cos ( nb j ) p j ( n ∈ Z ).Proof : Since c n = P n ( c ), where P n denotes the n -th Tchebishev polynomial, A is the closed unital subalgebra generated by c and the map χ → χ ( c ) is abijection from c A onto spec A ( c ). Now let χ ∈ c A . The sequence ( χ ( c n )) n ≥ is ascalar cosine sequence, and we have sup n ≥ ¯¯ cos ( na ) − χ ( c n ) ¯¯ < spec A ( c ) : = © λ = χ ( c ) : χ ∈ c A ª isfinite. Hence c A is finite. Let χ , . . . , χ m be the elements of c A . It follows from thestandard one-variable holomorphic functional calculus, se for example [9], thatthere exists for every j ≤ m an idempotent p j of A such that χ j ( p j ) = χ k ( p j ) = k j . Hence p j p k = j k , and m P j = p j is the unit element of A . Let x ∈ A . Then ( p j c n ) n ∈ Z is a cosine sequence in the commutative unitalBanach algebra p j A , and spec p j A ( p j c ) = { χ j ( c )}.Since sup n ≥ °° p j cos ( na ) − p j c n °° ≤ k p j k , the sequence ( p j c n ) n ≥ is boun-ded, and it follows from theorem 2.3 that ( p j c n ) n ≥ is a scalar sequence, andthere exists β j ∈ [0, π ] such that p j c n = χ j ( c n ) p j = cos ( n β j ) p j for n ∈ Z .7ence c n = m P j = χ j ( c n ) p j = m P j = cos ( n β j ) p j for n ≥
1. Since A is the closedsubalgebra of A generated by c , we have x = m P j = χ j ( x ) p j for every x ∈ A , whichshows that A is isomorphic to C m . ä Corollary 2.4.
Let a ≥ ∈ R , and let k ( a ) be the largest positive real number msuch that Γ ( a , m ) = { a } for every m < k ( a ). If ( C ( n )) n ∈ Z is a cosine sequence ina Banach algebra A such that sup n ≥ k C ( n ) − cos ( na )1 A k < k ( a ), then C ( n ) = cos ( na )1 A for n ∈ Z .Theorem 2.3 does not extend to cosine families over general abelian groups,as shown by the following easy result. Proposition 2.5.
Let G : = ( Z /3 Z ) N . Then there exists a G-cosine family ( C ( g )) g ∈ G with values in l ∞ which satisfies the two following conditions(i) sup g ∈ G k l ∞ − C ( g ) k = , (ii) The algebra generated by the family ( C ( g )) g ∈ G is dense in l ∞ .Proof : Elements g of G can be written under the form g = ( g m ) m ≥ , where g m ∈ {0, 1, 2}. Set C ( g ) : = µ cos µ g m 𠶶 m ≥ .Then ( C ( g )) g ∈ G is a G -cosine family with values in l ∞ which obviously satis-fies (i) since cos ¡ π ¢ = cos ¡ π ¢ = − .Now let φ = ( φ m ) m ∈ Z be an idempotent of l ∞ , and let S : = { m ≥ | φ m = g m = m ∈ S , g m = m ≥ m ∉ S , and set g = ( g m ) m ≥ . We have C (0 G ) − C ( g ) = l ∞ − C ( g ) = φ ,and so φ ∈ A . We can identify l ∞ to C ( β N ), the algebra of continuous func-tions on the Stone-C˘ech compactification of N , and β N is an extremely discon-nected compact set, which means that the closure of every open set is open, seefor example [1], chap. 6, sec. 6. Since the characteristic function of every openand closed subset of β N is an idempotent of l ∞ , the idempotents of l ∞ separatepoints of β N , and it follows from the Stone-Weierstrass theorem that A is densein l ∞ , which proves (ii). ä The values of the constant k ( a ) It was shown in [17] that k (0) = . We also have the following result. Proposition 3.1.
We have k ( a ) = p if a π is irrrational, and k ( a ) < p if a π isrational . Proof : Assume that a π ∉ Q . Then 3 a ∉ ± a + π Z , and we have k ( a ) ≤ sup n ≥ | cos ( na ) − cos (3 na ) | = sup x ∈ R | cos ( x ) − cos (3 x ) | = p b π in Q , then sup n ≥ | cos ( na ) − cos ( nb ) | =
2, and wealso have sup n ≥ | cos ( na ) − cos ( nb ) | = p a − qb ∉ π Z for ( p , q ) (0, 0). So ifsup n ≥ | cos ( na ) − cos ( nb ) | <
2, there exists p ∈ Z \ {0}, q ∈ Z \ {0} and r ∈ Z suchthat p a − qb = r π .If p
6= ± q then it follows from lemma 3.5 of [10] that we havesup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ | cos ( nq a ) − cos ( nqb ) | = sup n ≥ | cos ( qna ) − cos ( pna ) |= sup x ∈ R | cos ( q x ) − cos ( p x ) | = sup x ∈ R ¯¯¯¯ cos µ pq x ¶ − cos ( x ) ¯¯¯¯ ≥ p b = ± a + s π r , where r ∈ Z \ { −
1, 0, 1}, and wecan restrict attention to the case where b = a + s π r where r ≥
2, 1 ≤ s ≤ r − g cd ( r , s ) =
1. It follows from Bezout’s identity that there exists for every p ≥ u ∈ Z such that ub − ua − p π r ∈ π Z . If r is even, set p = r . We have, sincethe set © e i (2 n + a ª n ≥ is dense in the unit circle, sup n ≥ | cos ( nb ) − cos ( na ) | = sup n ∈ Z | cos ( nb ) − cos ( na ) |≥ sup n ≥ | cos ((2 n + ub ) − cos ((2 n + ua ) |= sup n ≥ | cos ((2 n + ua ) | = r is odd, and set p = r − . We have sup n ≥ | cos ( nb ) − cos ( na ) | ≥ sup n ≥ | cos ((2 n + ub ) − cos ((2 n + ua ) |≥ sup n ≥ ¯¯¯ cos ((2 nr + ua ) − cos ³ (2 nr + ua + (2 nr + ³ π − π r ´´¯¯¯ sup x ∈ R ¯¯¯ cos ( x ) + cos ³ x − π r ´¯¯¯ ≥ cos ³ π r ´ ≥ p > p a π is rational. If the order of a is equal to 1, then k ( a ) = a equals 2 or 4.Otherwise we have k ( a ) ≤ sup n ≥ | cos ( na ) − cos (3 na ) | = max ≤ n ≤ u | cos ( na ) − cos (3 na ) | .We have | cos ( nx ) − cos (3 nx ) | < π p if x ∉ ± ar ccos ³ p ´ + π Z . If na ∈ ± ar ccos ³ p ´ + π Z for some n ≥
1, then arccos ³ p ´ π would be rational, and α : = p + p i p wouldbe a root of unity. So β = α = − + p i would have the form β = e ik π n for some n ≤ k ≥ n such that g cd ( k , n ) = Q ( β ) be the smallest subfield of C containing Q ∪ β . Since 3 β + β + =
0, the degree of Q ( β ) over Q is equal to 2. On the other hand the Galoisgroup G al ( Q ( β )/ Q ) is isomorphic to ( Z / n Z ) × , the group of invertible elementsof Z / n Z , and we have, see [21], theorem 2.5 H ( n ) = d e g ( Q ( β )/ Q ) = H ( n ) = c ar d (( Z / n Z ) × ) denotes the number of integers p ∈ {1, . . . , n }such that g cd ( p , n ) = P ( n ) be the set of prime divisors of n . It is weil-known that we have, wri-ting n = Π p ∈ P ( n ) p α p , see for example [21], exercise 1.1, H ( n ) = Π p ∈ P ( n ) p α p − ( p − H ( n ) = n = n =
4, and n =
6. Since β β
1, and β
1, we seethat βπ is irrational, and so k ( a ) < p if a π is rational. ää We know that if a π is rational, and if b π is irrational, then sup n ≥ | cos ( na ) − cos ( nb ) | =
2. We discuss now the case where a π and b π are both rational, with b ∉ ± a + π Z . Lemma 3.2.
Let a , b ∈ (0, π ]. (i) If a ≤ b ≤ π , or if π ≤ b ≤ π , with ¯¯ b − π ¯¯ ≥ a , thensup n ≥ | cos ( na ) − cos ( nb ) | > (ii) If π ≤ b ≤ π , and if b ≥ a , then os ( a ) − cos ( b ) > a ≤ b ≤ π , let p be the largest integer such that pb < π , and set q = p +
1. We have π ≤ qb ≤ π , 0 ≤ q a ≤ π , and we obtain sup n ≥ | cos ( na ) − cos ( nb ) | ≥ cos ¡ q a ¢ − cos ¡ qb ¢ ≥ cos µ π ¶ + cos ³ π ´ > π ≤ b ≤ π , with | b − π | ≥ a , and set c = | b − π | . Since ¯¯ b − π ¯¯ ≤ π , we have 21 a ≤ c ≤ π , and we obtain sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ | cos (3 na ) − cos (3 nb ) |= sup n ≥ | cos (3 na ) − cos ( nc ) | > π ≤ b ≤ π , and if b ≥ a , then 0 < a ≤ π , and we have cos ( a ) − cos ( b ) ≥ cos ³ π ´ + cos ³ π ´ > Lemma 3.3.
Let p , q be two positive integers such that p < q . (i) If q p , then there exists u p , q ≥ such that, if or d ( a ) ≥ u p , q we havesup n ≥ | cos ( np a ) − cos ( nq a ) | > p (ii) If q = p , then for every m < p there exists u p ( m ) ≥ such that if or d ( a ) ≥ u ( m ) we have sup n ≥ | cos ( np a ) − cos (3 np a ) | > m .Proof : Set λ = sup x ∈ R | cos ( p x ) − cos ( q x ) | = sup x ≥ | cos ( p x ) − cos ( q x ) | . Anelementary verification shows that λ > p if q p , and λ = p if q = p , seefor example [10]. Now let µ < λ , and let η < δ be two real numbers such that | cos ( p x ) − cos ( q x ) | > µ for η ≤ x ≤ δ . Since { e i an } n ≥ = { e ni π u } ≤ n ≤ u , we see that sup n ≥ | cos ( np a ) − cos ( nq a ) | > µ if π u < δ − η , and the lemma follows. ä Lemma 3.4.
Assume that a π and b π are rational, let u ≥ be the order of a and letv be the order of b . (i) If u v , u v , v u then sup n ≥ | cos ( na ) − cos ( nb ) | ≥ + cos ¡ π ¢ > > p . 11 ii) If u = v , and if b ∉ ± a + π Z , then there exists w ∈ Z such that ≤ w ≤ u and g cd ( u , w ) = satisfyingsup n ≥ cos( na ) − cos ( nb ) | = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ nw π u ¶¯¯¯¯ . (2) Conversely if a ∈ π Q has order u , then for every integer w such that g cd ( w , u ) = there exists b ∈ π Q of order u satisfying (2).(iii) If v = u , then there exists an integer w such that ≤ w ≤ u and g cd ( u , w ) = satisfyingsup n ≥ cos( na ) − cos ( nb ) | = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ nw π u ¶¯¯¯¯ . (3) Conversely if a ∈ π Q has order u , then for every integer w such that g cd ( w , u ) = there exists b ∈ π Q of order u satisfying (3).(iv) If u = v , then there exists an integer w such that ≤ w ≤ u and g cd ¡ u , w ¢ = satisfyingsup n ≥ cos( na ) − cos ( nb ) | = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ nw π u ¶¯¯¯¯ . (4) Conversely if the order u of a ∈ π Q is divisible by then for every integer wsuch that g cd ¡ u , w ¢ = there exists b ∈ π Q of order u satisfying (4). Proof : (i) Assume that u v , say, u < v , and let w ub ,which is a divisor of v . We have ub = πα w , with g cd ( α , w ) =
1, and there exists γ ≥ αγ − ∈ w Z . We obtain sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ | cos ( nu γ a ) − cos ( nu γ b ) | = sup ≤ n ≤ w − cos µ n π w ¶ .If w is even, then sup n ≥ | cos ( na ) − cos ( nb ) | =
2. If w is odd, set s = w − . Weobtain sup n ≥ | cos ( na ) − cos ( nb ) | ≥ − cos µ s π w ¶ = + cos ³ π w ´ .If w ≥
5, we obtain sup n ≥ | cos ( na ) − cos ( nb ) | ≥ + cos ³ π ´ > > p w =
3, let d = g cd ( u , v ), and set r = ud . Then w = = vd > r . So either r = r =
2. 12f r =
2, we have u = d , v = d , a = p π d = p π d with p odd, b = q π d with g cd ( q , 3 d ) =
1, and we obtain sup n ≥ | cos ( na ) − cos ( nb ) | = | cos (3 d a ) − cos (3 d b ) |≥ | cos (3 p π ) − cos (2 q π ) | = r = u = d and v = d = u .We thus see that if v > u and v u , then sup n ≥ | cos ( na ) − cos ( nb ) | ≥ + cos ¡ π ¢ > > p , which proves (i).(ii) Assume that u = v , and that b ∉ ± a + π Z . There exists α , β ∈ {1, . . . , u − α β , α u − β such that a ∈ ± απ u + π Z and b ∈ ± βπ u + π Z , and g cd ( α , u ) = g cd ( β , u ) =
1. It follows from Bezout’s identity that there exists γ ∈ Z such that αγ − ∈ u Z . If βγ ± ∈ u Z then we would have αβγ ± α ∈ α u Z ⊂ u Z , and β ± α ∈ u Z , which is impossible. Hence γβ − w ∈ u Z for some w ∈ {2, . . . , u − g cd ( w , u ) = g cd ( γ , u ) = g cd ( β , u ) =
1, and we have sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ | cos( n γ a ) − cos ( n γ b ) |= sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ nw π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n απ u ¶ − cos µ n α w π u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n απ u ¶ − cos µ n βπ u ¶¯¯¯¯ = sup n ≥ | cos ( na ) − cos ( nb ) | .By replacing w by u − w if necessary, we can assume that 2 ≤ w ≤ u .Now let w ∈ Z such that g cd ( u , w ) =
1. We have a = απ u , with g cd ( α , u ) = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ nw π u ¶¯¯¯¯ = sup n ≥ cos( na ) − cos ( nb ) | ,with b = w απ u , which has order u .(iii) Now assume that v = u . There exists α ∈ {1, . . . , u −
1} and β ∈ {1, . . . , 3 u −
1} such that a ∈ ± απ u + π Z and b ∈ ± βπ u + π Z , and g cd ( α , u ) = g cd ( β , 3 u ) = γ ∈ Z such that βγ − ∈ u Z . Then g cd ( γ , 3 u ) =
1, and a fortiori g cd ( γ , u ) = w ∈ Z such that αγ ∈ ± w + u Z , and we see as above that we have sup n ≥ | cos ( na ) − cos ( nb ) | = sup n ≥ ¯¯¯¯ cos µ n απ u ¶ − cos µ n βπ u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n αγπ u ¶ − cos µ n βγπ u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ nw π u ¶ − cos µ n π u ¶¯¯¯¯ .13onversely let a = απ u ∈ π Q have order u , and let w ∈ Z be such that g cd ( u , w ) =
1. If α is not divisible by 3, then g cd ( α , 3 u ) =
1. If α is divisible by 3, then u is notdivisible by 3, and so α + u ∈ α + u Z is not divisible by 3. So we can assume wi-thout loss of generality that α is not divisible by 3, and there exists β ≥ αβ − ∈ u π Z . Similarly we can assume without loss of generality that w isnot divisible by 3, and there exists γ ≥ w γ − ∈ u π Z . Set b = αγπ u .Then b has order 3 u , and we see as above that we have sup n ≥ ¯¯¯¯ cos µ nw π u ¶ − cos µ n π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n αγ w π u ¶ − cos µ n αγπ u ¶¯¯¯¯ = sup n ≥ | cos ( na ) − cos ( nb ) | ≥ sup n ≥ ¯¯¯¯ cos µ n αγ w β w π u ¶ − cos µ n αγβ w π u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ nw π u ¶ − cos µ n π u ¶¯¯¯¯ ,which concludes the proof of (iii).(iv) Clearly, the first assertion of (iv) is a reformulation of the first assertionof (iii). Now assume that the order u of a ∈ π Q is divisible by 3, set v = u , write a = απ u , and let w ∈ Z such that g cd ( w , v ) =
1. We see as above that we canassume without loss of generality that g cd ( u , w ) = g cd ( α , u ) =
1, we have a fortiori g cd ( α , v ) =
1, so that g cd ( α w , v ) = b : = α wu has order v and we see as above that a , b , u and w satisfy (4). ä In order to use lemma 3.4, we introduce the following notions.
Definition 3.5.
Let u ≥ and denote by ∆ ( u ) the set of all integers s satisfying ≤ s ≤ u , g cd ( u , s ) = and let ∆ ( u ) = ∆ ( u ) \ {1}. We set σ ( u ) = i n f w ∈ ∆ ( u ) · sup n ≥ ¯¯¯¯ cos µ π u ¶ − cos µ w π u ¶¯¯¯¯¸ , θ ( u ) = i n f w ∈ ∆ ( u ) · sup n ≥ ¯¯¯¯ cos µ π u ¶ − cos µ w π u ¶¯¯¯¯¸ . with the convention θ ( u ) = if ∆ ( u ) = ; .Notice that ∆ ( u ) = ; if u =
2, 3, 4 or 6, and that ∆ ( u )
6= ; otherwise since aswe observed above H ( n ) = c ar d (( Z / n Z ) × ) ≥ n ∉ {1, 2, 3, 4, 6}.We obtain the following corollary, which shows in particular that the valueof k ( a ) depends only on the order of a . Corollary 3.6.
Let a ∈ π Q , and let u ≥ be the order of a . (i) If u is not divisible by then k ( a ) = i n f ( σ ( u ), θ ( u )). (ii) If u is divisible by then k ( a ) = i n f ( σ ¡ u ¢ , σ ( u ), θ ( u )).14roof : Set– Λ ( a ) = © b ∈ π Q | b ∉ ± a + π Z , or d ( b ) = or d ( a ) ª ,– Λ ( a ) = © b ∈ π Q | or d ( b ) = or d ( a ) ª ,– Λ ( a ) = © b ∈ π Q | or d ( b ) = or d ( a ) ª ,– Λ ( a ) = © b ∈ π Q | or d ( b ) or d ( a ) or d ( b ) ª ,and for 1 ≤ i ≤
4, set λ i ( a ) = i n f b ∈ Λ i ( a ) sup n ≥ | cos ( na ) − cos ( nb ) | ,with the convention λ i ( a ) = Λ i ( a ) = ; .Since b ∉ ± a + π Z if or d ( b ) or d ( a ), we have λ ( a ) ≤ p , and it followsfrom lemma 3.4(i) that we have k ( a ) = i n f ≤ i ≤ λ i ( a ) = i n f ≤ i ≤ λ i ( a ),and it follows from lemma 3.4 (ii), (iii) and (iv) that λ ( a ) = θ ( u ) if ∆ ( u )
6= ; ,that λ ( a ) = σ ( u ), and that λ ( a ) = σ ¡ u ¢ if u is divisible by 3. ä We have the following result.
Theorem 3.7.
Let m < p . Then the set Ω ( m ) : = { a ∈ [0, π ] : k ( a ) ≤ m } is finite. Proof : It follows from lemma 3.3 applied to π u and π u that there exists u ≥ u ≥ u ,( i ) sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ > m if 2 ≤ w ≤ i n f ³ u ´ ,( i i ) sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w + n π u ¶¯¯¯¯ > m if 0 ≤ w ≤ i i i ) sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w + n π u ¶¯¯¯¯ > m if 0 ≤ w ≤ u ≥ u , and let w be an integer such that 2 ≤ w ≤ u . Il w π u ≤ π /2, or if w π u ≥ π , it follows from lemma 3.2 and property (i) that we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ > m .Now assume that π ≤ w π u ≤ π . If ¯¯ w − u ¯¯ ≥
7, it follows from lemma 3.2 thatwe have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ > > m .15f ¯¯ w − u ¯¯ <
7, set r = | w − u | . Then 0 ≤ r ≤
20, and we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ nr π u ¶¯¯¯¯ .If u is not divisible by 3, then either r = s + r = s +
2, with 0 ≤ w ≤ sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ > m .If u is divisible by 3 then r is also divisible by 3. Set v = u and s = r . Then0 ≤ s ≤
6, and we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n π v ¶ − cos µ ns π v ¶¯¯¯¯ .If s ∈ {2, 3, 4, 5, 6} it follows from (i) that we have, if u ≥ u , sup n ≥ ¯¯¯¯ cos µ n π v ¶ − cos µ sn π u ¶¯¯¯¯ > m .Now assume that s =
0. If u ≥
15, then v ≥
5, and we have sup n ≥ ¯¯¯¯ cos µ n π v ¶ − cos µ sn π u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n π v ¶ − ¯¯¯¯ ≥ + cos ³ π ´ > > m .Now assume that s =
1. We have, with ǫ = ± sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n π v ¶ − cos µ n π v + n ǫπ ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n + π v ¶ − cos µ n + π v + ǫπ ¶¯¯¯¯ = p ¯¯¯¯ si n µ n π v + π v + ǫπ ¶¯¯¯¯ .There exists p ≥ q ∈ Z such that π − π v ≤ p π v + π v + ǫπ + q π ≤ π + π v ,and we obtain, for u ≥ w = v ± sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ ≥ p cos ³ π v ´ ≥ p cos ³ π ´ ≥ > m .We thus see that if u ≥ u is not divisible par 3, or if u ≥ max (21, 3 u ) isdivisible by 3, we have, for 2 ≤ w ≤ u , 16 up n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ w n π u ¶¯¯¯¯ ≥ m .It follows then from corollary 3.6 that if the order u of a ∈ [0, 2 π ] satisfies u ≥ max (21, 3 u ), we have k ( a ) > m . ä We now want to identify the real numbers a for which k ( a ) ≤ a ∈ π Q has order 1, 2 or 4, then sup n ≥ | cos ( an ) − cos (3 an ) | =
0. We alsohave the following elementary facts.
Lemma 3.8.
Let a ∈ π Q , and let u ∉ {1, 2, 4} be the order of a .
1. If u ∉ {3, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 22, 24, 30} thensup n ≥ | cos ( an ) − cos (3 an ) | >
2. If u ∈ {3, 6, 9, 12, 15, 18, 24, 30}, thensup n ≥ | cos ( an ) − cos (3 an ) | =
3. If u ∈ {5, 10}, then sup n ≥ | cos ( an ) − cos (3 an ) | = p
52 .
4. If u ∈ {8, 16}, then sup n ≥ | cos ( an ) − cos (3 an ) | = p
5. If u ∈ {11, 22}, thensup n ≥ | cos ( an ) − cos (3 an ) | = − cos µ π ¶ + cos µ π ¶ = cos µ π ¶ + cos µ π ¶ ≈ e i an } n ≥ = { e in π u } ≤ n ≤ u , and so we have sup n ≥ | cos ( an ) − cos (3 an ) | = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ n π u ¶¯¯¯¯ = sup ≤ n ≤ u ¯¯¯¯ cos µ n π u ¶ − cos µ n π u ¶¯¯¯¯ ,and the value of sup n ≥ | cos ( an ) − cos (3 an ) | depends only on the order u of a .17he function x → cos ( x ) − cos (3 x ) is increasing on h ar ccos ³ p ´i and de-creasing on h ar ccos ³ p ´ , − ar ccos ³ p ´i , and 0.275 π < ar ccos ³ p ´ < π .Since cos ( x ) − cos (3 x ) > x = π or if x = π , there exists a closedinterval I of length 0.058 π on which cos ( x ) − cos (3 x ) > u ≥ > ,there exists n ≥ n π u ∈ I , and we have sup n ≥ | cos ( an ) − cos (3 an ) | > ∀ n ≥ sup ≤ n ≤ u ¯¯ cos ¡ n π u ¢ − cos ¡ n π u ¢¯¯ for 3 ≤ u ≤
34 which are left to the reader. ä We now wish to obtain similar estimates for sup n ≥ ¯¯ cos ¡ π n ¢ − cos ¡ s π n ¢¯¯ for s ∈ {2, 4, 5, 6}. Set f s ( x ) = cos ( x ) − cos ( sx ), θ s = sup x ≥ | f ( s ) | , δ s = sup x ≥ | f "( s ) | .We have θ s = s is even, and a computer verification shows that θ s > s =
5. It follows from the Taylor-Lagrange inequality that if f s attains it maximumat α s , then we have, ¯¯ f s ( x ) − θ s ¯¯ ≤ δ s ¯¯ ( x − α s ) ¯¯ , ¯¯ f s ( x ) ¯¯ ≥ θ s − δ s ¯¯ ( x − α s ) ¯¯ ,and so | f s ( x ) | > ¯¯ ( x − α s ) ¯¯ ≤ θ s − δ s . So if l s < q θ s − δ s there exists a closedinterval of length 2 l s on which | f s ( x ) | > u s ≥ π l s be an integer. We obtain sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ > ∀ u ≥ u s . (5)Values for u s are given by the following table. s θ s δ s l s u s ≤ ≤
17 0.2425 135 > ≤
26 0.1519 216 2 ≤
37 0.1644 20We obtain the following result
Lemma 3.9.
Let u ≥ be an integer, and let s ≤ u be a nonnegative integer.If s then we havesup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ ns π u ¶¯¯¯¯ > s =
0, then 18 up n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ ns π u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − ¯¯¯¯ > s ≥
7, the result follows from lemma 3.2 (i). If s ∈ {2, 4, 6}, the result followsfrom the table since u ≥ s . If s =
5, the result also follows from the table for u ≥
21, and a direct computation shows that we have sup n ≥ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = sup ≤ n ≤ ¯¯¯ cos ³ n π ´ − cos ³ n π ´¯¯¯ = + cos ³ π ´ > ä Now set g s ( x ) = cos (3 x ) − cos ( sx ), θ s = sup x ≥ | g ( s ) | , δ s = sup x ≥ | g "( s ) | . Wehave θ s = s is even, and a computer verification shows that θ s > s = θ s > s = s = θ s > s = s = θ s > s =
19. We seeas above that if l s < q θ s − δ s , and if u s ≥ π l s is an integer, we have sup n ≥ ¯¯¯¯ cos µ sn π u ¶ − cos µ n π u ¶¯¯¯¯ > ∀ u ≥ u s . (6)We have the following table. s θ s δ s l s u s ≤
13 0.2774 124 2 ≤
23 0.2085 165 > ≤
34 0.1435 227 > ≤
58 0.1189 278 2 ≤
73 0.1170 2710 2 ≤
109 0.0958 3311 > ≤
130 0.0794 4013 > ≤
178 0.0727 4414 2 ≤
205 0.0698 4516 2 ≤
275 0.0603 5317 > ≤
298 0.0562 5619 > ≤
390 0.0486 6520 2 ≤
409 0.0494 64We will be interested here to the case where u is not divisible by 3 and where s π u ≤ π , which means that u ≥ s . So we are left with s = u =
8, 10 or 11, andwith s = u =
20. We obtain, by direct computations19 up n ≥ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos ³ n π ´ − cos µ n π ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = cos µ π ¶ − cos µ π ¶ = cos µ π ¶ + cos µ π ¶ ≈ sup n ≥ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos ³ n π ´ − cos µ n π ¶¯¯¯¯ > Lemma 3.10.
Let u , s be positive integers satisfying u ≥ u ≤ s ≤ u , with s ≥ so that u ≥ We havesup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = cos ¡ π ¢ + cos ¡ π ¢ if u = s = or if u = s = = p if u = s = or if u = s = = cos ¡ π ¢ + cos ¡ π ¢ if u = s = or if u = s = = if u = s = > otherwise .Proof : Set r = | s − u | . Since π − π = π − π = π , we have 0 ≤ π ru ≤ π . If r ≥
21, it follows from lemma 3.1(i) that sup n ≥ ¯¯ cos ¡ n π u ¢ − cos ¡ sn π u ¢¯¯ > u is not divisible by 3, then r is not divisible by 3 either, and it follows fromthe discussion above that if r r
2, we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ r n π u ¶¯¯¯¯ > r = ¯¯ s − u ¯¯ = . We saw above that in this situation sup n ≥ ¯¯ cos ¡ n π u ¢ − cos ¡ sn π u ¢¯¯ > u =
11, which gives s =
3, and wehave sup n ≥ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = sup ≤ n ≤ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ ¯¯¯¯ cos µ π ¶ + cos µ π ¶¯¯¯¯ = cos µ π ¶ + cos µ π ¶ ≈ r = ¯¯ s − u ¯¯ = , which gives s = u − if u ≡ s = u + if u ≡ sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ n π u ¶¯¯¯¯ .Since we must have ¯¯ s − u ¯¯ = , it follows from lemma 3.8 that if n ∉ {5, 8, 10, 11, 16, 22},or if u = s
2, or if u = s
3, or if u = s
3, or if u = s
4, or if u = s
5, or if u = s sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ > sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = sup ≤ n ≤ u ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = cos ¡ π ¢ + cos ¡ π ¢ if u = s =
2, or if u = s = p u = s =
3, or if u = s = cos ¡ π ¢ + cos ¡ π ¢ if u = s = u = s = u = v is divisible by 3. Then r is also divi-sible by 3. If r =
0, and if u
9, then we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n π v ¶ − ¯¯¯¯ > u =
9, then s =
3, and we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = sup ≤ n ≤ ¯¯¯¯ cos µ n π ¶ − cos µ n π ¶¯¯¯¯ = r =
3, which means that s = v + ǫ , with ǫ = ±
1. We have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = sup ≤ n ≤ v ¯¯¯¯ cos µ n π v ¶ − cos µ n π + ǫ n π v ¶¯¯¯¯ = sup ≤ n ≤ v ¯¯¯¯ sin µ n π + (1 + ǫ ) n π v ¶¯¯¯¯ ¯¯¯¯ si n µ − n π + (1 − ǫ ) n π v ¶¯¯¯¯ sup ≤ n ≤ v ¯¯¯ si n ³ n π ´¯¯¯ ¯¯¯¯ si n µ n π + n π v ¶¯¯¯¯ ≥ p sup ≤ n ≤ v ¯¯¯¯ si n µ (3 n + π + n + π v ¶¯¯¯¯ = p sup ≤ n ≤ v ¯¯¯¯ si n µ n π v + ( v + π v ¶¯¯¯¯ .Since si n ( x ) > p for π < x < π , there exists n ∈ {1, . . . , v } such that si n ³ n π v + ( v + π v ´ > p if v ≥
7, and we obtain sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ > u ≥ u = v = s = u = v = s = u = v = s = u = v = s = u = v = s = s = u ≤ s ≤ u is not satisfied for u = s = u = s = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ > u =
15 and s = > u =
18 and s = s = > u =
15 and s = > u =
12 and s = sup n ≥ ¯¯ cos ¡ n π u ¢ − cos ¡ sn π u ¢¯¯ > u ≤ s ≤ u when u is divisible by 3and when s − u ∈ { −
1, 0, 1}, unless u =
12 and s =
3. If u =
12 and s =
3, we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = sup n ≥ ¯¯¯ cos ³ n π ´ − cos ³ n π ´¯¯¯ = u = v is divisible by 3, and that 2 ≤ | s − v | ≤
6. Set again r = | s − u | , and set p = r , so that 2 ≤ p ≤
6. Notice also that p ≤ u since r ≤ u ,so that u ≥
24 and v ≥
8. We have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ ≥ sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ r n π u ¶¯¯¯¯ = sup n ≥ ¯¯¯¯ cos µ n π v ¶ − cos µ pn π u ¶¯¯¯¯ .It follows then from lemma 3.9 that sup n ≥ ¯¯ cos ¡ n π u ¢ − cos ¡ sn π u ¢¯¯ > p p =
3, then u ≥
36, and so v ≥
12. Since s − v = ±
3, it follows from lemma3.5 that we only have to consider the following cases :22 u = s = u = s =
12 or 18,– u = s =
15 or 21,– u = s =
21 or 27,– u = s =
27 or 33.Direct computations which are left to the reader show that we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ > u =
36 and s =
9, or if u =
45 and s =
12 or 18,or if u =
72 and s =
27, or if u =
90 and s =
27 or 33, > u =
54 and s = > u =
72 and s = > u =
36 and s = > u =
54 and s = ä Lemma 3.11.
Let u , s be positive integers satisfying u ≤ s ≤ u , with s ≥ so thatu ≥ We havesup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = ½ = if u = and s = > otherwise Proof : If s ≥
4, it follows from lemma 3.2(ii) that we have sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ > s = u = s = u = sup n ≥ ¯¯¯¯ cos µ n π u ¶ − cos µ sn π u ¶¯¯¯¯ = u = s = = u = s = = cos ¡ π ¢ + cos ¡ π ¢ ≈ u = s = ä We consider again the numbers θ ( u ) and σ ( u ) introduced in definition 3.6.It follows from lemma 3.8, lemma 3.9, lemma 3.10 and lemma 3.11 that wehave the following results. Lemma 3.12.
We have θ (5) = θ (10) = cos ¡ π ¢ + cos ¡ π ¢ , θ (8) = θ (16) = p θ (11) = θ (22) = cos ¡ π ¢ + cos ¡ π ¢ , and θ ( u ) > for u ≥ u u u u u u
22. 23 emma 3.13.
We have σ ( u ) = if u ∈ {1, 2, 3, 4, 5, 6, 8, 10} , σ ( u ) > otherwise. Hence if u is divisible by 3, we have σ ¡ u ¢ = u ∈ {3, 6, 9, 12, 15, 18, 24, 30} , σ ( u ) > Ω (1.5) = { a ∈ [0, π ] | k ( a ) ≤ Theorem 3.14.
Let a ∈ [0, π ]. – If a ∈ © π , π , π , π ª , then k ( a ) = cos ¡ π ¢ + cos ¡ π ¢ ≈
1, 1180. – If a ∈ © π , π , π , π , π , π ª , then k ( a ) = p ≈
1, 4142. – If a ∈ © π , π , π , π , π , π , π , π , π , π ª , then k ( a ) = cos ¡ π ¢ + cos ¡ π ¢ ≈
1, 4961. – If a ∈ © π , π , π , π , π ª ∪ © π , π , π , π , π , π ª ∪ © π , π , π ª ∪ © π , π , π , π , π , π , π , π ª , then k ( a ) = – For all other values of a , we have < k ( a ) ≤ p ≈ Corollary 3.15.
Let G be an abelian group, and let ( C ( g )) g ∈ G be a G-cosine fa-mily in a unital Banach algebra A such that sup g ∈ G k C ( g ) − c ( g ) k < p for somebounded scalar G-cosine family ( c ( g )) g ∈ G . Then C ( g ) = c ( g ) for every g ∈ G .Proof : Let g ∈ G . Since the scalar cosine sequence ( c ( ng )) n ∈ Z is bounded, astandard argument shows that there exists a ( g ) ∈ R such that c ( ng ) = cos ( na ( g ))1 A for n ∈ Z . Since k ( a ( g )) ≥ p , it follows from corollary 2.4 that C ( ng ) = cos ( na ( g ))1 A = c ( ng ) for n ∈ Z , and C ( g ) = c ( g ). ä Références [1] A.V. Arkhangel’skii and V.I. Ponomarev, Fundamentals of General Topology :Problems and Exercises, D.Reidel Publishing Company/Hindustan Publi-shing Company, 1984 (translated from Russian).[2] 1. W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued La-place Transforms and Cauchy Problems, Birkh¨auser, Basel, 2001.[3] A.Batkai, K.-J. Engel and M. Haase,
Cosine families generated by second or-der differential operators on W (0, 1) with generalized Wentzell boundaryconditions , Appl. Analysis 84 (2005), 867-876.[4] A. Bobrowski and W. Chojnacki,
Isolated points of some sets of bounded co-sine families, bounded semigroups, and bounded groups on a Banach space ,Studia Math. 217 (2013), 219-241.[5] A. Bobrowski, W. Chojnacki and A. Gregosiewicz,
On close-to-scalar one-parameter cosine families , submitted.246] W. Chojnacki,
Group decompositions of bounded cosine sequences , StudiaMath. 181 (2007), 61-85.[7] W. Chojnacki,
On cosine families close to scalar cosine families , J. AustralianMath. Soc. (2015), http ://dx.doi.org.10.1017/S1446788715000038[8] R. H. Cox,
Matrices all of whose powers lie close to the identity , Amer. Math.Monthly 73 (1966), 813.[9] H.G. Dales, Banach Algebras and Automatic Continuity, London Mathema-tical Society Monographs 24, Clarendon Press, Oxford, 2001.[10] J. Esterle,
Bounded cosine functions close to continuous scalar bounded co-sine functions , submitted, arXiv :1502.00150v2[11] M. Haase,
The functional calculus approach to cosine operator functions .Recent Trends in Analysis. Proceedings of the Conference in honour of N.K.Nikolski held in Bordeaux 2011, Theta Foundation 2013, 123-147.[12] M. Haase,
The group reduction for bounded cosine functions on UMDspaces , Math. Zeit. 262 (2) (2009), 281-299.[13] R. A. Hirschfeld,
On semi-groups in Banach algebras close to the identity ,Proc. Japan Acad. 44 (1968), 755.[14] J.P. Kahane and R. Salem, Ensembles parfaits et séries trigonométriques,Hermann, Paris, 1963.[15] B. Nagy,
Cosine operator functions and the abstract Cauchy problem,
Period.Math. Hungar. 7 (1976), 15-18.[16] M. Nakamura and M. Yoshida,
On a generalization of a theorem of Cox ,Proc. Japan Acad. 43 (1967), 108-110.[17] F. Schwenninger and H. Zwart,
Zero-two law for cosine families , J. Evol. Equ.(2015), http ://dx.doi.org.10.1007/s00028-015-0272-8[18] F. Schwenninger and H. Zwart,
Less than one, implies zero , arXiv :1310.6202.[19] C Travis and G. Webb,
Cosine families and abstract nonlinear second orderdifferential equation , Acta Math. Acad. Sci. Hungar 32(1978), 75-96.[20] L. J. Wallen,