Cauchy-Davenport type theorems for semigroups

Let $\mathbb{A} = (A, +)$ be a (possibly non-commutative) semigroup. For $Z \subseteq A$ we define $Z^\times := Z \cap \mathbb A^\times$, where $\mathbb A^\times$ is the set of the units of $\mathbb{A}$, and $$\gamma(Z) := \sup_{z_0 \in Z^\times} \inf_{z_0 \ne z \in Z} {\rm ord}(z - z_0).$$ The paper investigates some properties of $\gamma(\cdot)$ and shows the following extension of the Cauchy-Davenport theorem: If $\mathbb A$ is cancellative and $X, Y \subseteq A$, then $$|X+Y| \ge \min(\gamma(X+Y),|X| + |Y| - 1).$$ This implies a generalization of Kemperman's inequality for torsion-free groups and strengthens another extension of the Cauchy-Davenport theorem, where $\mathbb{A}$ is a group and $\gamma(X+Y)$ in the above is replaced by the infimum of $|S|$ as $S$ ranges over the non-trivial subgroups of $\mathbb{A}$ (Hamidoune-K\'arolyi theorem).Comment: To appear in Mathematika (12 pages, no figures; the paper is a sequel of arXiv:1210.4203v4; shortened comments and proofs in Sections 3 and 4; refined the statement of Conjecture 6 and added a note in proof at the end of Section 6 to mention that the conjecture is true at least in another non-trivial case

Upper and lower densities have the strong Darboux property

Let $\mathcal{P}({\bf N})$ be the power set of $\bf N$. An upper density (on $\bf N$) is a non\-decreasing and subadditive function $\mu^\ast: \mathcal{P}({\bf N})\to\bf R$ such that $\mu^\ast({\bf N}) = 1$ and $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \subseteq \bf N$ and $h,k \in {\bf N}^+$, where $k \cdot X + h := \{kx + h: x \in X\}$. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper P\'olya, and upper analytic densities are examples of upper densities. We show that every upper density $\mu^\ast$ has the strong Darboux property, and so does the associated lower density, where a function $f: \mathcal P({\bf N}) \to \bf R$ is said to have the strong Darboux property if, whenever $X \subseteq Y \subseteq \bf N$ and $a \in [f(X),f(Y)]$, there is a set $A$ such that $X\subseteq A\subseteq Y$ and $f(A)=a$. In fact, we prove the above under the assumption that the monotonicity of $\mu^\ast$ is relaxed to the weaker condition that $\mu^\ast(X) \le 1$ for every $X \subseteq \bf N$.Comment: 10 pages, no figures. Fixed minor details and streamlined the exposition. To appear in Journal of Number Theor

On the notions of upper and lower density

Let $\mathcal{P}({\bf N})$ be the power set of ${\bf N}$. We say that a function $\mu^\ast: \mathcal{P}({\bf N}) \to \bf R$ is an upper density if, for all $X,Y\subseteq{\bf N}$ and $h, k\in{\bf N}^+$, the following hold: (F1) $\mu^\ast({\bf N}) = 1$; (F2) $\mu^\ast(X) \le \mu^\ast(Y)$ if $X \subseteq Y$; (F3) $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$; (F4) $\mu^\ast(k\cdot X) = \frac{1}{k} \mu^\ast(X)$, where $k \cdot X:=\{kx: x \in X\}$; (F5) $\mu^\ast(X + h) = \mu^\ast(X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Polya, and upper analytic densities, together with all upper $\alpha$-densities (with $\alpha$ a real parameter $\ge -1$), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (F1)-(F5), and we investigate various properties of upper densities (and related functions) under the assumption that (F2) is replaced by the weaker condition that $\mu^\ast(X)\le 1$ for every $X\subseteq{\bf N}$. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.Comment: 26 pp, no figs. Added a 'Note added in proof' at the end of Sect. 7 to answer Question 6. Final version to appear in Proc. Edinb. Math. Soc. (the paper is a prequel of arXiv:1510.07473

On the commutation of generalized means on probability spaces

Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$. We say that $(f,g)$ is a $(\lambda, \mu)$-switch if, for every $\mathscr{L} \otimes \mathscr{M}$-measurable function $h: X \times Y \to \mathbf{R}$ for which $h[X\times Y]$ is contained in a compact subset of $I$, it holds $$f^{-1}\!\left(\int_X f\!\left(g^{-1}\!\left(\int_Y g \circ h\;d\mu\right)\right)d \lambda\right)\! = g^{-1}\!\left(\int_Y g\!\left(f^{-1}\!\left(\int_X f \circ h\;d\lambda\right)\right)d \mu\right)\!,$$ where $f^{-1}$ is the inverse of the corestriction of $f$ to $f[I]$, and similarly for $g^{-1}$. We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that $(f,g)$ is a $(\lambda, \mu)$-switch if and only if $f = ag + b$ for some $a,b \in \mathbf R$, $a \ne 0$.Comment: 9 pages, no figures. Fixed minor details. Final version to appear in Indagationes Mathematica