Maximal deviation of large powers in the Nottingham group

1 Introduction

Let 𝔽p be the field of order p, for a fixed prime p. Let 𝒩 be the Nottingham group over 𝔽p which is the group of all formal power series f⁢(x)∈𝔽p⁢[[x]] with leading term x, under formal substitution: given g,f∈𝒩 set (f⁢g)⁢(x)=f⁢(g⁢(x)). For brevity, f is used instead of f⁢(x) if it is not confusing. Consider the following chain of open subgroups of 𝒩:

The depth of f∈𝒩∖{1}, D⁢(f), is the integer k≥1 such that f∈𝒩k∖𝒩k+1, while D⁢(1)=∞. The distance of f and g is usually defined by

which makes 𝒩 an ultrametric space. It is clear that d⁢(f,g)=d⁢(fj,gj) if p∤j. So in understanding the distance of powers the crucial case is when the exponent j is a power of p.

The following definition is by Keating [3].

He defined this constant to prove the following theorem [3, Corollary 2] and to show that for m=1 the bound is sharp [3, Theorem 1 (b)].

He conjectured that the bound was sharp for every m>1 as well. In this paper we confirm his conjecture except for the case p=2, k=1, when it is not sharp for m>2.

For p=2, k=1 we have D⁢(g2m⁢f-2m)≥n+2m+2-11+e⁢(k,n) which is strictly stronger than the bound of Theorem 1.1 if m>2. This and the exact bounds for the various values of n are treated fully in [1].

Theorems 1.1 and 1.2 have the following metric version.

Crucial in the proof is the understanding of the Engel words. The general formulas (see Lemma 2.3 and Lemma 2.4), albeit aesthetic, are not sufficient for direct constructions. But reinterpreting them in linear algebraic terms allows direct computation of the leading terms of [a,s⁢(p-1)b] for arbitrary s≥0, see the formulation in Corollary 3.4.

The choice for f,g in Theorem 1.2 provides controllably increasing sequences D⁢(f)<D⁢(fp)<D⁢(fp2)<⋯ and D⁢(g)<D⁢(gp)<D⁢(gp2)<⋯. In fact, these are the slowest possible, see Lemma 2.2.

2 Preliminaries

First we take a closer look at the commutators in the Nottingham group. Let a,b∈𝒩. The commutator of a and b is [a,b]=[a,1b]=a-1b-1ab. The Engel word [a,mb] is defined inductively as [a,mb]=[[a,m-1b],b] for m≥2.

For later use we state the following two simple corollaries of Lemma 2.1, which are well known and fundamental for the Nottingham group. The first was noted for ε⁢(x) in the proof, the second follows from the first:

By [2, Lemma 1], D⁢(gp)≥p⁢D⁢(g) if gp≠1. Combined with (2.2) we get the form we will use.

Theorem 1.1 is, in fact, a generalisation of this lemma.

The next two results will be used in the proof of the main theorem but they are also of independent interest. The first is a generalisation of Lemma 2.1. Note that it also holds when b=1, k=∞.

Now we will describe Keating’s constructions and what we will need from his results. Let b,u∈𝒩 such that

where β⁢(x)=rk+rk+1⁢x+rk+2⁢x2+⋯, and

Assume p∤k. For s≤k, h≥1 we construct matrices Ah:=Ah,n,s∈Ms×s⁢(𝔽p). This matrix Ah is a linear operator that takes s-many coefficients of [u,h-1b] starting from x(h-1)⁢k+n+j+1 and gives s-many coefficients of [u,hb] starting from xh⁢k+n+j+1. This matrix is constructed formally using formula (2.9), below. The product Πp-1:=A1⁢A2⁢⋯⁢Ap-1 will be our main concern. It is an operator that takes s-many “leading” coefficients of u and gives s-many “leading” coefficients of [u,p-1b].

Define w1=u and wh+1=[wh,b] for h≥0. Following [3], we determine the coefficients of wh+1 from those of

where θ⁢(x)=t0+t1⁢x+t2⁢x2+⋯:

(2.6)

It follows from the expansion above that for 0≤j≤k-1 the coefficient of xh⁢k+n+j+1 in wh+1 is

using the abbreviation

For s≤k and h≥1 we define the matrix Ah=Ah,n,s∈Ms×s⁢(𝔽p) as

Clearly Ah depends only on s, b and the remainder of n modulo p.

Let now νh∈𝔽ps be the row vector whose entries are the coefficients of x(h-1)⁢k+n+j+1 in wh for 0≤j≤s-1. It follows from (2.7) that

For h≥1 define a matrix Πh=Πh,n,s∈Ms×s⁢(𝔽p) by setting Πh=A1⁢A2⁢⋯⁢Ah. Then we have νp=ν1⁢Πp-1.

Recall now the definition of k0 and e⁢(k,n) from Definition 1. Keating introduced the matrices above for s=e⁢(k,n)+1, and we follow his notation. He went on to show that in his e⁢(k,n)+1×e⁢(k,n)+1 matrix Πp-1 the first e⁢(k,n) columns are 0 (see [3, Cases 3–4]). (NB: He claimed this only for n>(p-1)⁢k+p, because he used it only in that case, however it works also for arbitrary n≥k+k0.) Using our extended notation, we may restate his result as follows:

From this we instantly get:

In the rest of the paper our concern will be this top right-hand corner block, T.

3 Constructions

Lemma 4 of [3] can be generalized as the following lemma, the proof being almost identical.

Consider a fixed k. By the definition of e⁢(k,n), the sum n+e⁢(k,n) is constant in the intervals k+t⁢p≤n≤k+t⁢p+k0 (with t≥0). Lemma 3.1 allows us to assume that k+k0+t⁢p≤n<k+(t+1)⁢p. If n=k+k0+t⁢p, then e⁢(k,n)=0, otherwise e⁢(k,n)=k0. We are going to assume this in the proof of Theorem 1.2, below.

Let p∤k and k+k0+t⁢p≤n≤k+(t+1)⁢p for some nonnegative integer t. We define

This describes the number of significant digits for the following definition. For u∈𝒩n let μn⁢(u)∈𝔽pe′⁢(k,n)+1 denote the vector consisting of the first e′⁢(k,n)+1 coefficients of u.

For given n,k as above and λ∈𝔽p we define an e′⁢(k,n)+1×e′⁢(k,n)+1 matrix Tn⁢[λ] as

and

and Tn⁢[λ]=(λ) if k+k0+p⁢t<n<k+p⁢(t+1).