On Some Diophantine Inequalities Involving the Exponential Function

1965 ◽  
Vol 17 ◽  
pp. 616-626 ◽  
Author(s):  
A. Baker
Keyword(s):  
Real Number ◽  
Exponential Function ◽  
Diophantine Inequalities ◽  
Real Numbers ◽  
Image Position ◽  
Positive Integers

It is well known that for any real number θ there are infinitely many positive integers n such thatHere ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

Banach games

1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Perfect Information ◽  
Real Numbers ◽  
The Real ◽  
Infinite Game ◽  
Image Position ◽  
The Relationship

Abstract.Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.

scholarly journals A Generalisation of a Result of Abel with an Application to Tree Enumerations

1965 ◽  
Vol 14 (3) ◽  
pp. 239-241 ◽  
Author(s):  
M. M. Robertson
Keyword(s):  
Real Numbers ◽  
Image Position ◽  
Positive Integers

We prove the following theorem, which was established by Abel (1) for the case u = 1.Theorem. If u, k are positive integers and x, 1, , u, are real numbers, thenwhere the sum is taken over all distinct ordered solutions (s1, , su) in non-negative integers of the equation .

On the Number of Primitive Lattice Points in a Parallelogram

1953 ◽  
Vol 5 ◽  
pp. 456-459 ◽  
Author(s):  
Theodor Estermann
Keyword(s):  
Real Number ◽  
Preceding Paper ◽  
Lattice Points ◽  
Primitive Lattice Points ◽  
Image Position ◽  
Positive Integers

1. Let a be any irrational real number, and let F(u) denote the number of those positive integers for which (n, [nα]) = 1. Watson proved in the preceding paper that

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals A Cyclic Inequality and an Extension of it. II

1962 ◽  
Vol 13 (2) ◽  
pp. 143-152 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Real Numbers ◽  
Positive Real ◽  
Cyclic Inequality ◽  
Image Position ◽  
Positive Integers

Throughout this paper, unless otherwise stated, n and L stand for positive integers and α, t, x, x1, x2, … for positive real numbers. Letwhereand

The definition of measure in function space

1950 ◽  
Vol 46 (1) ◽  
pp. 19-27
Author(s):  
H. R. Pitt
Keyword(s):  
Function Space ◽  
Real Variable ◽  
Fundamental Result ◽  
Real Numbers ◽  
Real Functions ◽  
Definition Of ◽  
Image Position ◽  
The Right ◽  
Positive Integers

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).

Some metrical theorems in diophantine approximation

1951 ◽  
Vol 47 (1) ◽  
pp. 18-21 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Infinitely Many Solutions ◽  
Real Numbers ◽  
Integer Solutions ◽  
The Difference ◽  
Image Position

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequalityhas infinitely many integer solutions q > 0. In particular, if α is any real number, the inequalityhas infinitely many solutions.

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