Bound on prime power divisors of denominators in Baker-Campbell-Hausdorff formula

History
These bounds were discovered by Lazard in his paper on the Lazard correspondence.

Statement
For a prime $$p$$ and a natural number $$n$$, denote by $$f(p,n)$$ the largest $$k$$ such that, if we truncate the Baker-Campbell-Hausdorff formula to terms that involve products of length at most $$n$$, then one or more of the denominators is divisible by $$p^k$$. Then, we have:

$$f(p,n) \le \left[\frac{n - 1}{p - 1}\right]$$

where $$[]$$ denotes the greatest integer function.

Corollaries

 * One particular fact that follows immediately from this is that the Baker-Campbell-Hausdorff formula for terms of degree less than $$p$$ does not contain any denominator divisible by $$p$$. This is crucial to the establishment of the Lazard correspondence.

Facts used

 * 1) uses::Bound on prime power divisors in associative formula equals bound on prime power divisors in Lie formula for Lie element

Proof for associative version of formula
The proof basically flows along the steps used to obtain the Baker-Campbell-Hausdorff formula in deducing the Baker-Campbell-Hausdorff formula from associative algebra manipulation. Consider the ring of formal power series over $$\mathbb{Q}$$ in the two non-commuting variables $$X$$ and $$Y$$.

Consider a $$p$$-adic valuation on $$\mathbb{Q}$$, i.e., a valuation $$v_p: \mathbb{Q} \setminus \{ 0 \} \to \mathbb{Z}$$ that sends a rational number $$a/b$$ to the integer $$k$$ such that $$a/(bp^k)$$ in reduced form has no divisor of $$p$$ for either the numerator or the denominator.

Extend the valuation to a value $$1/(p-1)$$ on the formal variables $$X,Y$$. The valuation can then be extended to the whole ring. We then use multiplicativity to compute the valuation at various terms:

$$\! v_p(X^n/n!)=n/(p-1) - v_p(n!) \geq 1/(p-1)$$

where we have used that $$v_p(n!) \leq [(n-1)/(p-1)]$$. Then, $$v_p(\exp X - 1)\geq 1/(p-1)$$. Denote $$W = \exp(X)\exp(Y) - 1$$. It is easy to see that $$v_p(W) \geq 1/(p-1)$$. The Baker-Campbell-Hausdorff formula is obtained by expanding

$$\! \log(1 + W)$$

Note that $$v_p(W^n/n)=nv_p(W)-v_p(n)\geq n/(p-1) - v_p(n!) \geq 1/(p-1)$$. That is, $$v_p(\log(1 + W))\geq 1/(p-1)$$. For coefficients in degree $$n$$, we obtain:

$$v_p(\mbox{coeff in degree n}) \geq 1/(p-1) - n/(p-1)=-(n-1)/(p-1)$$

Proof for Lie version of formula
This follows from the associative version and Fact (1).

Case $$p = 2$$
In the case $$p = 2$$, we obtain that:

$$f(2,n) \le n - 1$$.

We consider the first few cases:

Case $$p = 3$$
In the case $$p = 3$$, we obtain that:

$$f(3,n) \le [(n - 1)/2]$$.

We consider the first few cases: