Cyclotomic units modulo $p$-th powers in the cyclotomic tower Let $E$ be the unit group of the cyclotomic field $L_n=\mathbb{Q}(\zeta_{p^n})$ where $\zeta_{p^n}$ is a primitive $p^n$-th root of unity. Let $G=Gal(L_n/\mathbb{Q})$ and let $\Delta=Gal(L_1/\mathbb{Q})$. If we let $\omega: (\mathbb{Z}/p\mathbb{Z})^{\times} \to \mu_{p-1}\in \mathbb{Z}_p$ be the Teichmuller character (which we may also regard as a character to $\mathbb{Z}/p^n\mathbb{Z}$ by reducing modulo $p^n$), then we may embed $\Delta$ into $G$ by sending $\sigma_a \mapsto \sigma_{\omega(a)}$. In this way, $E$ is a $\Delta$-module and $E/E^p$ is a $\mathbb{Z}/p\mathbb{Z}[\Delta]$-module. Define the usual idempotents for $i=0,\dots,p-2$
\begin{align*}
\theta_i=-\sum_{a=1}^{p-1}a^i\sigma_{\omega(a)}^{-1}
\end{align*}
and decompose $E/E^p$ as a direct sum of its eigenspaces in the usual way. Now if we further suppose that $p$ is regular, then $p$ does not divide the index of the cyclotomic units in the full unit group so we may take cyclotomic units as generators for each of the subspaces. It is shown in Washington's book that when $n=1$, $\theta_i(E/E^p)=0$ for odd $i \neq 1$, $\theta_1(E/E^p)=\langle \zeta_p \rangle$, $\theta_0(E/E^p)=0$ and $\theta_i(E/E^p)\cong \mathbb{Z}/p\mathbb{Z}$ for even $i\neq 0$. In addition, if $g$  is a primitive root modulo $p$, we have that
\begin{align*}
E_i=\prod_{a=1}^{p-1}\Big(\zeta^{(1-g)/2}\frac{1-\zeta^g}{1-\zeta}\Big)^{a^i\sigma_a^{-1}}
\end{align*}
are generators for $\theta_i(E/E^p)$ and even $i$.
When $n>1$, things are not so simple.
The odd components are the same since every unit in $L_n$ is a product of a root of unity and a real unit and the odd components of real units vanish.
What I am having trouble with is determining the even components. For the $n=1$ case, there was a simple, perhaps naive, guess that turned out to be true: If $E_{+}$ is the group of real units, then $E_{+}/E_{+}^p\cong (\mathbb{Z}/p\mathbb{Z})^{(p-3)/2}$ by the Dirichlet unit theorem, the $0$-th component vanishes since $\theta_0$ is just the inverse of the norm, so the obvious guess is that the remaining $(p-3)/2$ even eigenspaces  are all cyclic of order $p$. For $n>1$, the $0$-th component doesn't necessarily vanish since it isn't a norm to $\mathbb{Q}$ but a norm to the subfield of degree $p^{n-1}$ over $\mathbb{Q}$. Also, $E_{+}/E_{+}^p \cong (\mathbb{Z}/p\mathbb{Z})^{p^{n-1}(p-1)/2-1}$ so unless I'm completely overlooking something it doesn't seem obvious how the sizes of the components should be distributed.
What I do have is a set of generators for $E_{+}/E_{+}^p$: If $g$ is a primitive root modulo $p^n$ then
\begin{align*}
\zeta_{p^n}^{(1-g)/2}\frac{1-\zeta_{p^n}^g}{1-\zeta_{p^n}}
\end{align*}
generates the  real cyclotomic units modulo $\{\pm 1\}$ over $\mathbb{Z}[G]$ so hitting all Galois conjugates of these elements with the idempotents $\theta_i$ will give us a set of generators for the components but it will be redundant.
Has anyone come across an analysis of this decomposition or something relevant?
 A: Perhaps you are not aware of the deep level of your question. A partial answer is known, in close relation with the so-called Iwasawa Main Conjecture (now a theorem of Mazur-Wiles (1984) when the base field is $\mathbf Q$, and Wiles (1990) when it is a totally real number field). Let us stick here to your hypotheses and notations but, for simplification, assume $p$ odd and consider only totally real fields and even parts (so + disappears from the indices). Given an abelian group $B$ of finite type, $B/pB$ is naturally an $\mathbf F_p$-vector space of finite dimension.
Recall that the subgroup $C_n$ of cyclotomic units inside $E_n$ has finite index equal to the class number $h_n$. Applying the snake lemma to the exact sequence $1 \to C_n \to E_n \to E_n/C_n \to 1$, we derive (with obvious notations) an exact sequence of vector spaces $ 1\to (E_n/C_n)[p] \to C_n/C_n^p \to E_n/E_n^p \to (E_n/C_n)/p \to 1$, so that $dim E_n/E_n^p= dim C_n/C_n^p$, and the problem is brought back to the study of $dim C_n/C_n^p$. This remains of course valid when working characterwise, i.e. applying idempotents of $\Delta$. This approach presents at the same time an advantage ($C_n$ is more or less "explicitly" known) and a drawback (the indirect intervention of the $p$-class-groups $A_n$, which  are the main arithmetic mystery). Note that the Vandiver conjecture is known to "propagate" along the cyclotomic tower, hence the  class-group problem vanishes in that case.
Without Vandiver, the point of view of Iwasawa's theory is to study the $A_n$ not one by one, but all together, more precisely to study $structurally$ the projective system $Y_{\infty}:=\varprojlim A_n$ w.r.t. norms. Actually $Y_{\infty}$ is endowed with a canonical structure of noetherian torsion module over the algebra $\Lambda:=\varprojlim \mathbf Z_p[\Gamma_n]$, where $\Gamma_n=Gal(L_{\infty}/\mathbf Q_n)$. The main technical point is that the seemingly complicated algebra $\Lambda$ is isomorphic to the complete power series algebra $\mathbf Z_p [[T]]$, which allows to attach to each noetherian torsion $\Lambda$-module $Z$ a characterisic power series $char_{\Lambda} (Z)$ describing the $\Gamma$-action on $Z$ analogously to the characteristic polynomial attached to a linear map acting on a vector space over a field. To make a lengthy story short, Iwasa introduced an important arithmetic $\Lambda$-module $X_{\infty}$ defined as the Galois group over $L_{\infty}$ of the maximal abelian pro-$p$-extension of $L_{\infty}$ which is unramified outside $p$, and he showed in a celebrated paper (1964) that his MC is equivalent to $char_{\Lambda}(X_{\infty})= char_{\Lambda} (E^1_{\infty}/ C^1_{\infty})$. For simplicity, I don't explain the upper indices $(.)^1$; suffice it to say that they stand for minor modifications of $E_{\infty}$ and $C_{\infty}$ (see the notations at the beginning of [CS]). This was the crucial step just before the final Mazur-Wiles proof of the MC, which expresses $char_{\Lambda}(X_{\infty})$ in terms of the $p$-adic zêta function (see thm. 1.4.3 of [CS]).
To get back to our original problem, let us recall that in the course of his theorem, Iwasawa showed that $C^1_{\infty}$ is $\Lambda $-cyclic, generated by the proj. lim. of an "explicit" system b of cyclotomic units (lemma 4.3 of [CS]). To "go down" to the $n$-th level, we have just to take the $\Gamma_n$ co-invariants of $C^1_{\infty}$ and b. Applying idempotents, we can get in priciple the semi-simple decomposition along $\Delta$ of $C_n/C_n^p$. "In principle" only, because the description of the co-invariants of b, although "explicit", is still not accurate enough without Vandiver. Actually, by the so-called "reflection theorems", the situation is analogous to the semi-simple decomposition of the $\mathbf F_p$-vector spaces $A_n/{A_n}^p$ .
Ref. Instead of Washington's book, in the particular case of the base field  $\mathbf Q$, I prefer the monograph [CS] by J. Coates and R. Sujatha, "Cyclotomic Fields and Zeta Values", Springer Mono. in Math., where the simplifications due to the base field allow to delineate much more clearly the flow of arguments.
