# Transcendence measure: of $\ln(a/b)$

In the book " Number Theory IV Transcendental Numbers" written by Parsin and Shafarevich (book, page 104) it is asserted that to explicit a transcendence measure of a complex number $$w$$, it is sufficient to prove that there is an infinite sequence of polynomials $$P_m(x)\in\mathbb Z[x]$$ of fixed degree such that $$0 where $$c$$, $$\lambda_1$$, and $$\lambda_2$$ are constants, such that $$c > 1, \lambda_l > \lambda_2 > 0$$ with $$H(P_m)=\max\{|\text{coefficients of P_m}|\}$$. I do not know why. Can anyone give the argument behind this assertion or a reference? This result is used to give a transcendence measure of $$\ln(r)$$ ($$r$$ is a rational)

• @fedor Why did you remove your post. It was very interesting, even it was not an answer to my question. Feb 18, 2023 at 20:49
• Did you ask essentially the same question before? Just use polynomials $P_m(z)^k$. By the way, that book is written by N. I. Fel'dman and Yu. V. Nesterenko, and not by Parshin and Shafarevich (who are editors of the series). Feb 19, 2023 at 2:15
• A detail answer would be welcome Feb 19, 2023 at 2:28
• It's not a research question. It's an exercise for students, who just saw a definition of transcendence measure. By the way, the correct claim is that you can get a bound for a transcendence measure (not the actual measure, that is not known). Feb 19, 2023 at 2:40

Let $$a/b$$ be an approximation of $$w$$ such that $$|w-a/b|=b^{-\kappa}$$. Then $$P_m(a/b)=P_m(w)+(w-a/b)P_m'(\theta)$$ for certain $$\theta$$ between $$a/b$$ and $$w$$. Note that $$P_m(a/b)$$ is either 0 or at least $$b^{-d}$$ in absolute value, where $$\deg P_m\leqslant d$$. Choose $$m$$ such that $$b^{-d}\geqslant 2e^{-\lambda_2 m}$$, say, $$m=\lceil \frac{\log 2+d\log b}{\lambda_2}\rceil$$.
Then, if $$|P_m(a/b)|\geqslant b^{-d}$$, we get $$C(d,w)\cdot c^m b^{-\kappa}\geqslant |b^{-\kappa}P_m'(\theta)|=|(w-a/b)P_m'(\theta)|=|P_m(a/b)-P_m(w)|\geqslant e^{-\lambda_2 m},$$ thus $$\kappa\log b\leqslant m(\log c+\lambda_2)+O(1)$$ and $$\kappa\leqslant d(1+\frac{\log c}{\lambda_2})+o(1)$$.
If $$P_m(a/b)=0$$, then analogously $$C(d,w)\cdot c^m b^{-\kappa}\geqslant |b^{-\kappa}P_m'(\theta)|=|(w-a/b)P_m'(\theta)|=|P_m(w)|\geqslant e^{-\lambda_1 m},$$ so $$\kappa\log b\leqslant m(\log c+\lambda_1)+O(1)$$, and $$\kappa\leqslant d\cdot \frac{\log c+\lambda_1}{\lambda_2}+o(1)$$.
So, in both cases we may conclude that the irrationality measure of $$w$$ does not exceed $$d\cdot \frac{\log c+\lambda_1}{\lambda_2}.$$