The simultaneous diophantine approximation of certain kth roots

1970 ◽  
Vol 67 (1) ◽  
pp. 75-86 ◽  
Author(s):  
Charles F. Osgood
Keyword(s):  
Diophantine Approximation ◽  
Simultaneous Approximation ◽  
Rational Numbers ◽  
Simultaneous Diophantine Approximation ◽  
Explicit Bounds ◽  
Positive Integers

In a recent paper in this journal(1) Baker obtained a result giving effectively computable bounds on the simultaneous approximation of certain numbers of the form (ab-1)nj/n by rational numbers of the formpιq-1 where a, b, n, the nι and q are positive integers while the pι are non-negative integers. In this paper we shall obtain explicit bounds on the simultaneous approximation of certain numbers of the form rβ1, …, rβn where β and each rι, 1 ≤ j ≤ n, are positive rational numbers.

scholarly journals Simultaneous diophantine approximation of rational numbers

1972 ◽  
Vol 22 (1) ◽  
pp. 1-9 ◽  
Author(s):  
T. Cusick
Keyword(s):  
Diophantine Approximation ◽  
Rational Numbers ◽  
Simultaneous Diophantine Approximation

scholarly journals SIMULTANEOUS DIOPHANTINE APPROXIMATION IN R2 × C × Qp

2013 ◽  
Vol 56 (1) ◽  
pp. 79-86
Author(s):  
Ella Kovalevskaya
Keyword(s):  
Diophantine Approximation ◽  
General Problem ◽  
Simultaneous Approximation ◽  
Simultaneous Diophantine Approximation ◽  
New Form

ABSTRACT An analogue of the convergence part of Khintchine’s theorem (1924) for simultaneous approximation of integral polynomials at the points (x1, x2, z,w) ∈ R2 × C × Qp is proved. It is a solution of the more general problem than Sprindźuk's problem (1980) in the ring of adeles. We use a new form of the essential and nonessential domain methods in metric theory of Diophantine approximation

Simultaneous Diophantine Approximation Using Primes

1988 ◽  
Vol 20 (4) ◽  
pp. 289-292 ◽  
Author(s):  
A. Balog ◽  
J. Friedlander
Keyword(s):  
Diophantine Approximation ◽  
Simultaneous Diophantine Approximation

scholarly journals KHINTCHINE'S THEOREM AND TRANSFERENCE PRINCIPLE FOR STAR BODIES

2006 ◽  
Vol 02 (03) ◽  
pp. 431-453
Author(s):  
M. M. DODSON ◽  
S. KRISTENSEN
Keyword(s):  
Distance Function ◽  
Diophantine Approximation ◽  
Direct Proof ◽  
Distance Functions ◽  
Transference Principle ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies

Analogues of Khintchine's Theorem in simultaneous Diophantine approximation in the plane are proved with the classical height replaced by fairly general planar distance functions or equivalently star bodies. Khintchine's transference principle is discussed for distance functions and a direct proof for the multiplicative version is given. A transference principle is also established for a different distance function.

scholarly journals Simultaneous diophantine approximation

1972 ◽  
Vol 6 (2) ◽  
pp. 317-318
Author(s):  
J.M. Mack
Keyword(s):  
Diophantine Approximation ◽  
Simultaneous Diophantine Approximation

scholarly journals On Simultaneous Diophantine Approximation in the Vector Space Q+Qα

2000 ◽  
Vol 82 (1) ◽  
pp. 12-24
Author(s):  
Edward B Burger
Keyword(s):  
Vector Space ◽  
Diophantine Approximation ◽  
Simultaneous Diophantine Approximation

scholarly journals Vertical Shift and Simultaneous Diophantine Approximation on Polynomial Curves

2014 ◽  
Vol 58 (1) ◽  
pp. 1-26
Author(s):  
Faustin Adiceam
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Rational Approximations ◽  
Vertical Shift ◽  
Polynomial Curves ◽  
Integer Coefficients ◽  
Simultaneous Diophantine Approximation ◽  
First Results

AbstractThe Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well-approximable points lying on a curve that is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.

scholarly journals Further Results on a Curious Arithmetic Function

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Long Chen ◽  
Kaimin Cheng ◽  
Tingting Wang
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Rational Numbers ◽  
Arithmetic Properties ◽  
Adic Valuation ◽  
Positive Integers

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

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