I am reading through the 2005 paper by Simons and Weger on the Collatz Conjecture.
Lemma 8 has the following conclusion:
$$\delta K < K +L < 1.000001\delta K$$
I am clear on most of the argument. I am clear on this point:
$$0 < K+L - \delta K < \frac{m}{X_0\log 2} \le \frac{K}{X_0\log 2} < 10^{-17}K$$
I can see how "the inequalities readily follow" but when I do my calculations, I am reaching a conclusion significantly better. I am wondering if I am making a mistake since it is not clear to me why the author would be reporting a weaker result.
When I add $\delta K$ to both sides, I get this:
$$\delta K < K + L < 10^{-17}K + \delta K = (10^{-17} + \delta)K < (6.31 \times 10^{-18} + 1)\delta K$$ which appears to be a better result than the conclusion.
Is my reasoning correct and the conclusion is provided to greatly simplify the result? Did I make a mistake in my reasoning?
If my reasoning is correct, can this result then be restated for example as:
$$\delta K < K + L < 1.0000000001\delta K$$
Your extra inequality of
$$(10^{-17} + \delta)K \lt (6.31 \times 10^{-18} + 1)\delta K \tag{1}\label{eq1A}$$
is, after dividing by $K$ and then subtracting $\delta$ from both sides, equivalent to
$$10^{-17} \lt 6.31 \times 10^{-18}\delta \iff \frac{10}{6.31} = 1.5847\ldots \lt \delta \tag{2}\label{eq2A}$$
Since the paper gives, near the top of page $53$ before its Lemma $1$, that
$$\delta = \frac{\log 3}{\log 2} = 1.5849\ldots \tag{3}\label{eq3A}$$
and assuming this definition is not changed anywhere later, which a scan of the paper seems to indicate this being the case, this means \eqref{eq2A} is true.
I don't know why the authors didn't use more decimal digits (e.g., up to $16$ zeroes before a $1$ instead of using just $5$ zeroes) to state a tighter upper bound, but possibly they felt there wasn't any need for the extra accuracy for their purposes.