J. W. S. Cassels, An Introduction to Diophantine Approximation, (Cambridge University Press, Cambridge), 168 pp. (1965).  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... assumptions involving regularity, growth and dissipativeness conditions, this result requires f; g to depend quasiperiodically on the rapid space variable z = x= Moreover the spatial frequencies involved in this quasiperiodicity will be required to satisfy a Diophantine condition in the sense of [Cas57]. For complete details and an explicit condition including some discussion see section 3, as well as the independent companion paper [FV00] Going beyond estimate (1.5) which only compares solutions starting from identical initial conditions, we also aim at a quantitative homogenization estimate ....

.... 2 IR for which there exist small positive c; such that the Diophantine condition (3.39) holds, for all k 2 ZZ l nf0g: Then it is known from Diophantine approximation theory, that D is a set of full Lebesgue measure in IR meas IR l (IR nD ) 0; 3. 40) see for example [Cas57]. So our Diophantine estimate (3.38) holds true for a set of frequences = 1 ; n ) of full Lebesgue measure in IR From estimate (3.38) we conclude jS ( j H C 0 (3.41) for all = 1; n; provided that the sum k2IR n0 j k j H 1 (3.42) converges. A ....

J.W.S. Cassels. An Introduction to Diophantine Approximation. Cambridge University Press, 1957.

....1 0 and 2. We conclude that if x; y and z are integers, not all zero, and X = maxfjxj; jyj; jzjg, then j x 1 y 2 z j c Gamma1 2 X Gamma 1 where c 2 = c 2 2 Gamma 1 2 2 2 Gamma and 1 = 2 Gamma2 2 Gamma . 8 Proof : This is just a special case of Theorem II, Chapter 5 of [4]. 2 Applying this result in our situation yields fi fi fi x 4 p N 4 Gamma 1 y 4 p N 4 1 z fi fi fi Gamma c Gamma2 1 2 Gamma2 X Gamma2 2 Delta 1 2 Gamma where X = maxfjxj; jyj; jzjg. Thus, we may conclude, from (3.3) and (3.4) that if xyz 6= 0 and X = jzj, then fi ....

J.W.S. Cassels. An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge, 1957.

.... b;p for any integers b 1; p 2 is transcendental. Remark. Though b;p has been introduced for (b; p) PRNG systems with b 1; gcd(b; p) 1, p an odd prime, the transcendency result is valid for any integer pair (b; p) with p 2 and b 1. Proof. The celebrated Roth theorem states [36] [13] that if jP=Q j 1=Q 2 admits of in nitely many rational solutions P=Q (i.e. if is approximable to degree 2 for some 0) then is transcendental. We show here that b;p is approximable to degree p . Fix a k and write b;p = P=Q X n k 1 p n b p n ; where gcd(P; Q) 1 ....

J. W. S. Cassels, An introduction to diophantine approximations, Cambridge Univ. Press, Cambridge, 1957.

....mg. Given real numbers 1 ; m ; 1 ; m , a nonhomogeneous diophantine approximation is a sequence of integers P 1 ; Pm ; Q such that Q 0 and for all j 2 [m] jQ j P j j j . Nonhomogeneous diophantine approximation need not exist. Theorem 1. 2 (Kronecker, see [Cas57, Lov86]) Let 1 ; m ; 1 ; m 2 R. Then exactly one of the following holds. For any 0 there are P 1 ; Pm ; Q such that Q 0 and for all j 2 [m] jQ j P j j j . There are integers a 1 ; am such that P a j j is an integer and P a j j is not an ....

J. Cassels. An Introduction to Diophantine Approximations. Cambridge University Press, Cambridge, 1957.

....also prove a theorem on linear forms, of the type (0:7) jx 0 x 1 1 Delta Delta Delta xm m j X Gamma 1 for x 0 ; xm integers, 1 ; m as in (0. 5) and X = max 0im jx i j satisfying X X 0 ( 1 ; 1 ; m ) Standard transference arguments (see e.g. Cassels [5]) ensure that (0.3) implies (0.7) with exponent 1 = m( Gamma 1) m( Gamma 1) provided 1 1= m Gamma 1) Our result, however, is somewhat stronger. These results have direct applications to Diophantine equations which we will address in x8 and x9. For example, they permit solution of ....

J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge, 1957.

....numbers # 1 , #m , # 1 , #m , a nonhomogeneous diophantine # approximation is a sequence of integers P 1 , Pm , Q such that Q 0 and for all j # [m] Q# j P j # j # #. Nonhomogeneous diophantine # approximation need not exist. Theorem 1. 2 (Kronecker, see [Cas57, Lov86]) Let # 1 , #m ; # 1 , #m # R. Then exactly one of the following holds. For any # 0 there are P 1 , Pm , Q such that Q 0 and for all j # [m] Q# j P j # j # #. There are integers a 1 , am such that # a j # j is an integer and # a j # ....

J. Cassels. An Introduction to Diophantine Approximations. Cambridge University Press, Cambridge, 1957.

....1. The proofs of Theorems 4.6 and 4.7 come from purely number theoretic arguments. The approximation of t by is a classical area of research in number theory which is referred to as inhomogeneous Diophantine approximation. A classical result in this direction is the theorem of Minkowski [C] which states that if t is not in the orbit of then kt p k 1= 4p) has in nitely many integer solutions p and the constant 1 4 can not be improved in general. Here k k is the standard distance on S 1 . In the spirit of Minkowski s theorem we consider the set A ; ft 2 S 1 : kt ....

.... = mod ) 3.1. Inhomogeneous Diophantine approximation. Remember that A ; ft : kt p k p for in nitely many p 2 Ng: Minkowski has shown that for t not in the orbit of the inequality kt p k 1= 4p) has in nitely many solutions and in general the constant 1=4 is optimal [C]. A simple Borel Cantelli argument tells us that directions which can be approximated better than in the statement of Minkowski s theorem have zero measure. Proposition 3.1. Suppose that 1 and 62 Q . The Lebesgue measure of A ; is 0 while dimB A ; 1. The set A ; is residual. ....

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J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge University Press, Cambridge, 1957.

....the number of actually needed qubits in Shor s algorithm to factor a given composite number N , where N is the product of two randomly chosen primes of equal size. Although our method easily extends to a wider class of randomly chosen modules, we will concentrate here for clarity to this special case. Moreover, from a practical point of view, this is the most interesting case. By exploiting the inherent probabilism of quantum computation we are able to substitute the continued fraction algorithm to find a certain unknown fraction by a simultaneous Diophantine approximation. While the ....

....composite number N , where N is the product of two randomly chosen primes of equal size. Although our method easily extends to a wider class of randomly chosen modules, we will concentrate here for clarity to this special case. Moreover, from a practical point of view, this is the most interesting case. By exploiting the inherent probabilism of quantum computation we are able to substitute the continued fraction algorithm to find a certain unknown fraction by a simultaneous Diophantine approximation. While the continued fraction algorithm is able to find a Diophantine approximation to a single ....

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J. W. S. Cassels, An Introduction to Diophantine Approximations, Cambridge University Press, Cambridge, 1957.

....M. Because the f k are homogeneous forms in x and y, the image of Z 2 will be uniformly distributed in L provided that the only set of integers s 0 ; s d 1 such that s 0 f 0 (x; y) s d 1 f d 1 (x; y) N x M y (22) with integer coecients M and N , is s 0 = s d 1 = 0 ([4], page 64) The a k ( are integral polynomials in of degree jk 1j, and since the degree of h q is d=2 1 (cf. 6) and (10) as long as d=2 1 k d=2, the powers are independent over Q, so the integrality condition must be imposed separately on each coecient. Writing, for all k and l a ....

....(dimension d) namely the case n i = 0; i = 0; q d 1. This corresponds to the unique xed point of at the origin. Proof of theorem 2. That the n are convex polyhedra corresponding to level sets of the vector eld is part of proposition 3.1, to be proved below. By de nition ([4], Chapter IV) uniform distribution implies the equivalence of density in Z 2 and normalized volume in for any nite collection of elementary sets (parallelepipeds with sides parallel to the coordinate axes) To extend this correspondence to the polyhedra n , we note that the boundary of ....

J. W. S. Cassels, An introduction to diophantine approximations, Cambridge University Press, Cambridge (1957).

....that k# n k bae n 8n 2 N. iii) # is a PV number, and = # Gammak for some integer k 0 and some number 2 Q(#) such that Tr(# j ) 2 Z, 0 j r Gamma 1) A proof of Theorem 3, which will be our main tool in the analysis of the Fourier module of a substitution tiling, can be found in (Cassels, 1957). THE DIFFRACTION PATTERN OF SELFSIMILAR TILINGS 15 We now recall that multiplication by # maps the translation module T into itself. With respect to a basis of T , this linear mapping is described by a matrix M with integral entries. Clearly, # is an eigenvalue of M , and therefore must be an ....

Cassels, J. W. S. (1957) An Introduction to Diophantine Approximation, Cambridge University Press (Cambridge).

....to an integer vector. The Diophantine constant for w is defined by C t (w) liminf m m t mw where m = max( m i ) A Diophantine frequency has C d (w) 0. The theory of simultaneous approximation of frequency vectors is not as complete as continued fraction theory; see, e.g. [10]. We adopt a Farey approximation technique proposed in [11] Begin with three resonances m 1 = 1,0,0) m 2 = 0,1,0) m 3 = 0,0,1) each corresponding to a plane in R 3 . The set of three resonances delineates a cone (the positive octant) denoted by the matrix M = m 1 ,m 2 ,m 3 ) t . To ....

Cassels, J. W. S. (1965)An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge.

....be the denominators of successive convergents of continued fractions of the arbitrary irrational number . Then the inequality kq n k 1 2q n 1 holds true. As usual , we denote here by the symbol ktk the distance from t to the nearest integer. Proof. See inequality (16) on p. 16 of the book [11]. Note that if the partial denominators of continued fractions of the number are bounded, then the inequalities q n q n 1 q n hold true, and if is an algebraic number, then it follows from Roth s theorem (see [11, p. 128] that q n q n 1 q 1 n . In addition, by virtue of the ....

Cassels, J., An Introduction to the Diophantine Approximation, Cambridge: Cambridge Univ. Press, 1957.

....B(z) f(x; y) xy 1 xy zg: Consider a second irrational number fi which is rationally independent of ff and 1. An inhomogeneous diophantine approximation is a rational numbers p q which satisfies jqj: jqff fi Gamma pj z: 7:2) 16 M. POLLICOTT AND M. YURI By a result of Minkowski [1], providing z 1 4 there are infinitely many solutions to (7.2) The inhomogeneous transformation T plays a role in this problem akin to that of the continued fraction transformation for homogeneous diophantine approximations. Let Y = f(z; w) 2 R 2 : 0 w 1; 0 w Gamma z 1g: We shall ....

J. Cassels, An introduction to Diophantine approximation, C.U.P., Cambridge, 1957.

....for T due to H.S. Dumas [D1] We rst recall the notations for diophantine vectors: for all K 0, s 2 R, let D(s; K) f 2 S n 1 j 8k 2 Z n n f0g ; j kj Kjkj s g : 2:6) We recall that 8K 0 ; 8s n 1 ; D(s; K) 2:7) which is a variant of a result due to Dirichlet, see [Ca] chapter I, Theorem VI) and that 8s n 1 ; d meas (D(s; K) c ) O(K) 2:8) Theorem D. For all n 2 N and s n 1, there exists C(n; s) 0 such that, for all K 0 and all 2 D(s; K) T ( r) C 00 (n; s) Kr s : 5 We refer to [D2] ChGa] for an application of this type of ....

J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge (Eng.) University Press, (1957).

....hyperbolic. Furthermore, we have to discuss the case jtr(M)j 2 where both eigenvalues of M must be real and, due to j det(M)j = 1, one, say, must lie strictly outside the interval [ Gamma1; 1] and the other strictly inside. Consequently, the larger eigenvalue is a Pisot Vijayaraghavan number [6]. To solve the question for the symmetry group here, we may employ Dirichlet s unit theorem. It is clear that is a quadratic irrational and real, and its algebraic conjugate (the other solution of the characteristic polynomial) is different from it. So, from Lemma 3 we know that the centralizer ....

J. S. W. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge (1965).

....We define homogenous linear forms L 1 ; L n by L j (x) t j A x, where A = A Gamma1 ) t . If P n j=1 w j L j (x) has integer coefficients, then P n j=1 w j t j 2 L and P n j=1 w j ff j = i P n j=1 w j t j j m 2 Z. It follows by Kroneckers approximation theorem (see e.g. [C], p. 58) that there is a v 2 Z d and an X 0 such that kL j (v) Gamma ff j kZ 1 8n ; j = 1; n and jv i j X; i = 1; d; where k Delta kZ denotes the distance to Z. With u = A v Gamma m 2 L we obtain kut j kZ 1= 8n) and ku mk kA k p dX. Hence (u) 1=2 ....

J.W.S. Cassels, `An introduction to Diophantine approximation', Cambridge University Press, Cambridge (1965).

.... Dumas [D1] We first recall the notations for diophantine vectors: for all K 0, s 2 R, let D(s; K) f 2 S n Gamma1 j 8k 2 Z n n f0g ; j Delta kj Kjkj Gammas g : 2:6) We recall that 8K 0 ; 8s n Gamma 1 ; D(s; K) 2:7) which is a variant of a result due to Dirichlet, see [Ca] chapter I, Theorem VI) and that 8s n Gamma 1 ; d Gamma meas (D(s; K) c ) O(K) 2:8) Theorem D. For all n 2 N and s n Gamma 1, there exists C(n; s) 0 such that, for all K 0 and all 2 D(s; K) T ( r) C 00 (n; s) Kr s : We refer to [D2] ChGa] for an application ....

J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge (Eng.) University Press, (1957).

.... ) is badly approximable, if there exists a constant c 0 such that ae(ff 1 x 1 : ff d Gamma1 x d Gamma1 ) c Gamma maxfjx 1 j; jx d Gamma1 jg Delta Gamma(d Gamma1) for all non zero (x 1 ; x d Gamma1 ) 2 Z d Gamma1 , with ae(t) min n2Z fjt Gamma njg, see [6, 5, 31] for more details. Theorem 3.4 If ff = ff 1 ; ff d Gamma1 ) is badly approximable, then the set fR ff n (B; N) N 1g is bounded in R . On the other hand, for almost all ff (with respect to Lebesgue measure) the function R ff n (B; N) is unbounded as N 1. The key to the proof of ....

J.W.S. Cassels, An Introduction to Diophantine Approximation (Cambridge University Press, 1957).

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J. W. S. Cassels, An Introduction to Diophantine Approximation, (Cambridge University Press, Cambridge), 168 pp. (1965).

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J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge Univ. Press, New York, 1957.

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J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts no. 45, Cambridge University Press, London, 1957.

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J.W.S. Cassels, An introduction to Diophantine approximation, Cambridge, Cambridge University Press, 1957.

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J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge (1957).

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J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, Cambridge (1957).

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J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tract 45, Cambridge University Press, 1965.

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