# Bound for the additive period length of certain Sprague-Grundy functions

Let $$\left( Y_x \right)_{x=0}^\infty$$ be a sequence of finite subsets of $$\mathbb{Z}$$, and let $$G : \mathbb{N}_0 \to \mathbb{N}_0$$ be a greedy permutation, defined by

$$G(x) = \operatorname{mex} \left( \left\{ G(x') | x' < x \right\} \cup Y_x \right) ,$$ where $$\operatorname{mex}$$ is the minimum excluded operator. Such permutations appear as Sprague-Grundy functions in the study of certain combinatorial games like Nim and Wythoff's Game.

Dress, Flammenkamp and Pink showed that if $$\left( Y_x \right)$$ is additively periodic, then $$G$$ also becomes additively periodic. That is, if there exists $$p \in \mathbb{N}_1$$ such that $$Y_{x+p} = Y_x + p$$ for all $$x \in \mathbb{N}_0$$, then there exists $$\overline{p} \in \mathbb{N}_1$$ such that $$G(x+ \overline{p} ) = G(x) + \overline{p}$$ for all sufficiently large $$x$$. A simple proof of this is given by Landman.

Question: What is the best bound for $$\overline{p}$$ in terms of $$p$$, if we know $$\overline{M} = \max_{x \in \mathbb{N}_0} (Y_x - x)$$ and $$\underline{M} = \min_{x \in \mathbb{N}_0} (Y_x - x)$$ ?

The proof by Landman gives $$\overline{p} \leq 2^{\overline{M} + \underline{M} + 2 } p$$, and the original proof gives $$\overline{p} \leq \binom{| \underline{M}| + \overline{M} + 2}{ \min \left( | \underline{M} - 1 |, \overline{M} + 1 \right) } p$$. I believe I can show that $$\overline{p} \leq K_{\underline{M},\overline{M} } \: p$$, where

$$K_{\underline{M} , \overline{M} } \to \exp \left( \sqrt{ \operatorname{Li}^{-1} \left( |\underline{M} | + \overline{M} \right) } \right) \quad \mathrm{as} \quad \underline{M} \to -\infty , \overline{M} \to \infty .$$ Here, $$\operatorname{Li}$$ is the logarithmic integral.

Is this a known result, and do improved bounds exist?

(Disclaimer: I am an amateur matematician, and I have no opportunities to ask other matematicians in person, which is why I'm asking here.)