Let epsilon>0, let a_1,…,a_n be positive integers, and let b_1,…,b_n be non-zero integers such that a_1^b_1… a_n^b_n≠ 1. Lang–Waldschmidt conjecture. There is a number C(epsilon) depending only on epsilon such that |a_1^b_1… a_n^b_n-1|≥ fracC(epsilon)max_j|b_j||b_1… b_n a_1… a_n|^1+epsilon. The conjecture predicts a strong archimedean lower bound for non-vanishing multiplicative linear forms in logarithms. It is presented as an open conjecture motivating the paper's discussion of effective Diophantine approximation.

The Lang–Waldschmidt conjecture on linear forms in logarithms

Let $\epsilon>0$ , let $a_1,\ldots,a_n$ be positive integers, and let $b_1,\ldots,b_n$ be non-zero integers such that $a_1^{b_1}\cdots a_n^{b_n}\ne 1$ . Lang–Waldschmidt conjecture. There is a number $C(\epsilon)$ depending only on $\epsilon$ such that

\left|a_1^{b_1}\cdots a_n^{b_n}-1\right|\ge \frac{C(\epsilon)\max_j|b_j|}{\left|b_1\cdots b_n a_1\cdots a_n\right|^{1+\epsilon}}.

The conjecture predicts a strong archimedean lower bound for non-vanishing multiplicative linear forms in logarithms. It is presented as an open conjecture motivating the paper's discussion of effective Diophantine approximation.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Lang--Waldschmidt conjecture on linear forms in logarithms
Let $a_1,\ldots,a_n,b_1,\ldots,b_n$ be non-zero integers and define
$\Lambda=\sum_i b_i\log a_i\ne0.$
Set
$A=\max_i|a_i|,\qquad B=\max_i|b_i|.$
Lang--Waldschmidt conjecture. There is an effectively computable constant $C_n>0$ , depending only on $n$ , such that
$\log|\Lambda|\geqslant-C_n(\log A+\log B).$
This is the Lang--Waldschmidt conjecture on Baker's logarithmic sum estimates, invoked in the source as a conditional tool for obtaining stronger estimates related to dyadic approximation. Its resolution status is not established by the supplied text.
source: Demi Allen, Sam Chow and Han Yu, “Dyadic Approximation in the Middle-Third Cantor Set”, arXiv:2005.09300 (2020).

Sources & referencesView supporting material

Primary source

Hector Pasten, “On the arithmetic case of Vojta's conjecture with truncated counting functions”, arXiv:2205.07841 (2022).

diophantine-approximation linear-forms-in-logarithms transcendence-theory

1 week ago

The Lang–Waldschmidt conjecture on linear forms in logarithms

Equivalent formulations 1

Lang--Waldschmidt conjecture on linear forms in logarithms

Primary source

Progress summary

Solutions 0