A lower bound for non-vanishing linear forms

1975 ◽  
pp. 49-62
Author(s):  
D. W. Masser
Keyword(s):  
Lower Bound ◽  
Linear Forms

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

2020 ◽  
Vol 102 (3) ◽  
pp. 479-489
Author(s):  
XIANG GAO ◽  
SHENGYOU WEN
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Transcendental Number ◽  
Linear Forms In Logarithms ◽  
Multiplicative Semigroup ◽  
Linear Forms ◽  
Coin Tossing

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

On Diophantine approximations of the solutions of q-functional equations

2002 ◽  
Vol 132 (3) ◽  
pp. 639-659 ◽  
Author(s):  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Number Fields ◽  
Growth Conditions ◽  
Linear Forms ◽  
Dimension Estimate ◽  
Linearly Independent ◽  
Diophantine Approximations ◽  
Image Position

Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

On the Product of Three Homogeneous Linear Forms. IV

1943 ◽  
Vol 39 (1) ◽  
pp. 1-21 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Linear Transformation ◽  
Linear Forms ◽  
Special Linear ◽  
Absolute Value ◽  
Cubic Equation ◽  
Image Position ◽  
Homogeneous Linear

Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound offor integral values of u, v, w, not all zero. I proved a few years ago (1) thatmore precisely, thatexcept when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving

scholarly journals Linear Dependence Of Powers Of Linear Forms

2015 ◽  
Vol 29 (1) ◽  
pp. 131-138
Author(s):  
Andrzej Sładek
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Dependence ◽  
Linear Forms ◽  
Powers Of Linear Forms

AbstractThe main goal of the paper is to examine the dimension of the vector space spanned by powers of linear forms. We also find a lower bound for the number of summands in the presentation of zero form as a sum of d-th powers of linear forms.

On the product of three homogeneous linear forms and indefinite ternary quadratic forms

1955 ◽  
Vol 248 (940) ◽  
pp. 73-96 ◽  
Keyword(s):  
Lower Bound ◽  
Quadratic Forms ◽  
Type A ◽  
Strong Type ◽  
Geometry Of Numbers ◽  
Linear Forms ◽  
Integer Coefficients ◽  
Ternary Quadratic Forms ◽  
Cubic Forms ◽  
Homogeneous Linear

Isolation theorems for the minima of factorizable homogeneous ternary cubic forms and of indefinite ternary quadratic forms of a new strong type are proved. The problems whether there exist such forms with positive minima other than multiples of forms with integer coefficients are shown to be equivalent to problems in the geometry of numbers of a superficially different type. A contribution is made to the study of the problem whether there exist real <j>, ijr such that x(f>x—y | y[rx — z | has a positive lower bound for all integers x > 0, y , z . The methods used have wide validity.

scholarly journals A lower bound for linear forms in logarithms

1980 ◽  
Vol 37 ◽  
pp. 257-283 ◽  
Author(s):  
Michel Waldschmidt
Keyword(s):  
Lower Bound ◽  
Linear Forms In Logarithms ◽  
Linear Forms
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