Simultaneous Diophantine Approximation
نویسنده
چکیده
Using a method suggested by E. S. Barnes, it is shown that the simultaneous inequalities r(p — arf < c, r(q — fir) < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals a and /3, provided that c > 1/2.6394. This improves an earlier result of Davenport, who shows that the same conclusion holds if c > 1/46"" = 1/2.6043 • • •.
منابع مشابه
Best Simultaneous Diophantine Approximations under a Constraint on the Denominator
We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximati...
متن کاملKnapsack Public Key Cryptosystems And
This paper presents and analyzes cryptanalytic attacks on knapsack public key cryptosystems that are based on ideas from Diophantine approximation. Shamir’s attack on the basic Merkle-Hellman knapsack cryptosystem is shown to depend on the existence of ‘‘unusually good’’ simultaneous Diophantine approximations to a vector constructed from the public key. This aspect of Shamir’s attack carries o...
متن کاملA higher-dimensional Kurzweil theorem for formal Laurent series over finite fields
In a recent paper, Kim and Nakada proved an analogue of Kurzweil’s theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.
متن کاملDiophantine Approximation with Arithmetic Functions, I
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
متن کاملSimultaneous inhomogeneous Diophantine approximation on manifolds
In 1998, Kleinbock & Margulis [KM98] established a conjecture of V.G. Sprindzuk in metrical Diophantine approximation (and indeed the stronger Baker-Sprindzuk conjecture). In essence the conjecture stated that the simultaneous homogeneous Diophantine exponent w0(x) = 1/n for almost every point x on a non-degenerate submanifold M of Rn. In this paper the simultaneous inhomogeneous analogue of Sp...
متن کاملDiophantine Exponents of Affine Subspaces: the Simultaneous Approximation Case
We apply nondivergence estimates for flows on homogeneous spaces to compute Diophantine exponents of affine subspaces of Rn and their nondegenerate submanifolds.
متن کامل