Let $\unicode[STIX]{x1D70C}\in (0,\infty ]$ be a real number. In this short note, we extend a recent result of Marques and Ramirez [‘On exceptional sets: the solution of a problem posed by K. Mahler’, Bull. Aust. Math. Soc.94 (2016), 15–19] by proving that any subset of $\overline{\mathbb{Q}}\cap B(0,\unicode[STIX]{x1D70C})$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many analytic transcendental functions with rational coefficients and radius of convergence $\unicode[STIX]{x1D70C}$. This solves the question posed by K. Mahler completely.

For $p\,>\,0$ and for a given set $E$ of type ${{G}_{\delta }}$ in the boundary of the unit disc $\partial \mathbb{D}$ we construct a holomorphic function $f\,\in \,\mathbb{O}\left( \mathbb{D} \right)$ such that

$${{\int_{\mathbb{D}\backslash \left[ 0,\,1 \right]E}{\left| f \right|}}^{p}}\,d{{\mathfrak{L}}^{2}}\,<\,\infty \,\text{and}\,E\,=\,{{E}^{p}}\left( f \right)\,=\,\{\,z\,\in \,\partial \mathbb{D}\,:\,\int _{0}^{1}\,{{\left| f\left( tz \right) \right|}^{p}}\,dt\,=\,\infty \}.$$

In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb{D}$.

Assume that $\Omega$ is a Hartogs domain in ${{\mathbb{C}}^{1+n}}$, defined as $\Omega =\left\{ \left( z,w \right)\,\in \,{{\mathbb{C}}^{1+n}}\,:\left| z \right|\,<\,\mu \left( w \right),w\,\in H \right\}$, where $H$ is an open set in ${{\mathbb{C}}^{n}}$ and $\mu$ is a continuous function with positive values in $H$ such that –ln $\mu$ is a strongly plurisubharmonic function in $H$. Let ${{\Omega }_{w}}=\Omega \cap \left( \mathbb{C}\times \left\{ w \right\} \right)$. For a given set $E$ contained in $H$ of the type ${{G}_{\delta }}$ we construct a holomorphic function $f\in \mathbb{O}\left( \Omega \right)$ such that

$$E=\left\{ w\in {{\mathbb{C}}^{n}}:\int\limits_{{{\Omega }_{w}}}{{{\left| f\left( \cdot ,w \right) \right|}^{2}}d{{\mathfrak{L}}^{2}}=\infty } \right\}.$$

Some results are presented relating to questions raised in a recent paper by Anderson, Hayman and Pommerenke regarding the size of the set of boundary points of the unit disc at which a univalent function has a prescribed radial growth.

AMS 2000 Mathematics subject classification: Primary 30C55

Estimates are established for the number of integers of size N, in intervals of size $N^\theta$, that fail to admit a representation as the sum of s cubes (s = 5, 6). Thereby it is shown that almost all such integers are represented in the proposed manner. When s = 5 one may take $\theta = 10/21$, and when s = 6 one may take any $\theta >17/63$. Similar such conclusions are also established for the related problem associated with the expected asymptotic formula.

We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by 9, and not exceeding $X$, that fail to have a representation as the sum of 7 cubes of prime numbers, is $O\left( {{X}^{23/36+\varepsilon }} \right)$. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O\left( {{X}^{11/36+\varepsilon }} \right)$.

Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N)$ denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number $\epsilon$, one has $E(N) \ll N^{13/30 + \epsilon}$, thereby exploiting effectively the 'excess' fourth square of a prime so as to improve the recent bound $E(N) \ll N^{13/15 + \epsilon}$ due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.

2000 Mathematical Subject Classification: 11P32, 11P05, 11P55.

We prove some new results on quasi-regular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)-elliptic part, diffusions on loop spaces, and certain Fleming- Viot processes.