Two sets $A,B$ of positive integers are called exact additive complements if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow 1$. For $A=\{a_1<a_2<\cdots \}$, let $A(x)$ denote the counting function of A and let $a^*(x)$ denote the largest element in $A\bigcap [1,x]$. Following the work of Ruzsa [‘Exact additive complements’, Quart. J. Math. 68 (2017) 227–235] and Chen and Fang [‘Additive complements with Narkiewicz’s condition’, Combinatorica 39 (2019), 813–823], we prove that, for exact additive complements $A,B$ with ${a_{n+1}}/ {na_n}\rightarrow \infty $,

$$ \begin{align*}A(x)B(x)-x\geqslant \frac{a^*(x)}{A(x)}+o\bigg(\frac{a^*(x)}{A(x)^2}\bigg) \quad\mbox{as } x\rightarrow +\infty.\end{align*} $$

We also construct exact additive complements $A,B$ with ${a_{n+1}}/{na_n}\rightarrow \infty $ such that

$$ \begin{align*}A(x)B(x)-x\leqslant \frac{a^*(x)}{A(x)}+(1+o(1))\bigg(\frac{a^*(x)}{A(x)^2}\bigg)\end{align*} $$

for infinitely many positive integers x.

We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space $L_{a}^{2}(\mathbb{D})$ and weighted Bergman spaces $L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$ when $\unicode[STIX]{x1D6FC}$ is a nonnegative integer.

We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ over 𝔽1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q=1 as expected from the infinite genus of C. Then, we develop a theory of functorial 𝔽1-schemes which reconciles the previous attempts by Soulé and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of -schemes to interpret conceptually the spectral realization of zeros of L-functions.

Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure R is an integral domain if and only if R is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra.