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In this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc. [ABSTRACT FROM AUTHOR]

The Lp$L_p$ Busemann–Petty problem for 00$\varepsilon >0$ small enough, if K is an Lp$L_p$ intersection body and L is a star body such that for any u∈Sn−1$u\in S^{n-1}$, the following inequality: ∥u∥IpK−p≤∥u∥IpL−p+ε$$\begin{equation*} \Vert u\Vert _{I_pK}^{-p}\le \Vert u\Vert _{I_pL}^{-p}+\varepsilon \end{equation*}$$implies V(K)n−pn≤V(L)n−pn+Cε,$$\begin{equation*} V(K)^{\frac{n-p}{n}}\le V(L)^{\frac{n-p}{n}}+C\varepsilon , \end{equation*}$$where ∥·∥IpK$\Vert \cdot \Vert _{I_pK}$ is the Minkowski functional of IpK$I_pK$, and C is a positive number depending only on p and n. Moreover, we also prove the linear stability and separation results for the negative answer of the Lp$L_p$ Busemann–Petty problem. [ABSTRACT FROM AUTHOR]

For an expansive real matrix M=ρ−1C0ρ−1$M= \def\eqcellsep{&}\begin{bmatrix} \rho ^{-1} & \mathcal {C}\\ 0& \rho ^{-1} \end{bmatrix}$ and a noncollinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t}$D=\lbrace (0,0)^t,(\alpha _1,\alpha _2)^t,(\beta _1,\beta _2)^t\rbrace$ with α2−2β2∉3Z$\alpha _2-2\beta _2\notin 3\mathbb {Z}$, let μM,D$\mu _{M,D}$ be the self‐affine measure defined by μM,D(·)=13∑d∈DμM,D(M(·)−d)$\mu _{M,D}(\cdot)=\frac{1}{3}\sum _{d\in D}\mu _{M,D}(M(\cdot)-d)$. In this paper, some necessary and sufficient conditions for L2(μM,D)$L^2(\mu _{M,D})$ contains an infinite orthogonal exponential set or μM,D$\mu _{M,D}$ to be a spectral measure are given. [ABSTRACT FROM AUTHOR]