Irrationality measure is an extension of the binary of whether a number is rational or irrational, that also includes transcendental numbers in its definition. For any number $L$, if the following holds infinitely often for integers $p_n,q_n$:
$$0<\left|\frac{p_n}{q_n}-L\right|<\frac{1}{q_n^{1+\delta}}$$
Then we define irrationality measure separately for fundamental constants as the highest possible $\delta$ for which this holds for the best possible choice of integer sequences $p_n,q_n$, and for PCFs we define this as the highest possible $\delta$ allowed by the PCF's convergents.
We want to add database support for irrationality measure. The constant table should be augmented with a delta column of real floating point numbers. If a constant is extended as a named_constant, try to find its (best known) irrationality measure in the literature and upload that. Otherwise, if a constant is augmented as a pcf_canonical_constant, estimate $\delta$ numerically from the convergents. For other constants this can stay null.
Also, add the ability to estimate $\delta$ to PCF.eval.
Irrationality measure is an extension of the binary of whether a number is rational or irrational, that also includes transcendental numbers in its definition. For any number$L$ , if the following holds infinitely often for integers $p_n,q_n$ :
$$0<\left|\frac{p_n}{q_n}-L\right|<\frac{1}{q_n^{1+\delta}}$$ $\delta$ for which this holds for the best possible choice of integer sequences $p_n,q_n$ , and for PCFs we define this as the highest possible $\delta$ allowed by the PCF's convergents.
Then we define irrationality measure separately for fundamental constants as the highest possible
We want to add database support for irrationality measure. The$\delta$ numerically from the convergents. For other constants this can stay
constanttable should be augmented with adeltacolumn ofrealfloating point numbers. If a constant is extended as anamed_constant, try to find its (best known) irrationality measure in the literature and upload that. Otherwise, if a constant is augmented as apcf_canonical_constant, estimatenull.Also, add the ability to estimate$\delta$ to PCF.eval.