Small-divisor slabs and the positive measure of the Diophantine set: the resonance covering has measure $O(\alpha )$

Let $n\ge 1$, $\tau >n-1$, $\alpha >0$ and $R>0$, and write $\mathrm{\ell }(k):=\mathrm{\mid }k{\mathrm{\mid }}_1$ for $k\in {\mathbb{Z}}^n$ and $C_R:=(-R,R{)}^n$. For every nonzero $k$ put $S_k:=\{\nu \in C_R:\mathrm{\mid }k\cdot \nu \mathrm{\mid }<\alpha \mathrm{\ell }(k{)}^{-\tau }\},\mathcal{R}:={\bigcup }_{k\in {\mathbb{Z}}^n\setminus \{0\}}{S}_k,$ and let $D:=C_R\setminus \mathcal{R}$. Then:

1. Slab bound. For every $k\mathrm{\ne }0$, $\mathrm{\mid }{S}_k\mathrm{\mid }\le 2^n{n}^{n\mathrm{/}2}{R}^{n-1}\alpha \mathrm{\ell }(k{)}^{-\tau -1}\mathrm{.}$
2. Resonant set. With $C_{n,\tau }:=2^n{n}^{n\mathrm{/}2}{4}^n{\sum }_{m\ge 1}{m}^{n-2-\tau }$ (a finite constant, since $\tau >n-1$), $\mathrm{\mid }\mathcal{R}\mathrm{\mid }\le C_{n,\tau }{R}^{n-1}\alpha \mathrm{.}$
3. Diophantine set. $D$ is measurable, is contained in the $(\alpha ,\tau )$-Diophantine set $\{\nu \in C_R:\mathrm{\mid }k\cdot \nu \mathrm{\mid }\ge \alpha \mathrm{\ell }(k{)}^{-\tau }\text{ for all }k\mathrm{\ne }0\}$, and $\mathrm{\mid }D\mathrm{\mid }\ge (2R{)}^n-C_{n,\tau }{R}^{n-1}\alpha ;$ consequently, for every $0<\alpha <{\alpha }_0(R,n,\tau ):=(2R{)}^n\mathrm{/}(C_{n,\tau }{R}^{n-1})$ the Diophantine set inside $C_R$ has positive measure.
4. Boundary layer. For every $0<\rho <R$ the same bounds hold with $R$ replaced by $R-\rho$, so the Diophantine set inside the closed cube $[-R+\rho ,R-\rho {]}^n$ has measure at least $(2(R-\rho ){)}^n-C_{n,\tau }(R-\rho {)}^{n-1}\alpha$; in particular frequencies at distance at least $\rho$ from the boundary of $C_R$ form a positive-measure Diophantine set once $\alpha$ is small enough.

All measures are $n$-dimensional Lebesgue measure (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$); the countable union and the re-summation over shells use the standard countable bookkeeping convention of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), which is the only choice principle involved.

• Why $\tau >n-1$. The exponent in the summed series is $n-2-\tau$, so convergence is exactly $\tau >n-1$; this is a summation phenomenon, not an arithmetic one, and no arithmetic property of the frequencies beyond the linear forms $k\cdot \nu$ is used.
• Uniformity in $\alpha$. The constant $C_{n,\tau }$ is independent of $\alpha$ and $R$, so the estimate is uniform over cubes; this is the form needed when the positive-measure set is compared with the label set of the KAM iteration.
• No sharper statement. The bound is an upper estimate for the union of slabs; no lower bound on the resonance measure and no claim about the topology of $D$ (in particular no claim that $D$ has interior) is made.