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

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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 1Other wordings

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).

References

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

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Lang--Waldschmidt conjecture on linear forms in logarithms

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