Counting carefree couples

A pair of natural numbers (a,b) such that a is both squarefree and coprime to b is called a carefree couple. A result conjectured by Manfred Schroeder (in his book `Number theory in science and communication') on carefree couples and a variant of it are established using standard arguments from elementary analytic number theory. Also a related conjecture of Schroeder on triples of integers that are pairwise coprime is proved.Comment: Updated version of 2005 update of 2000 version. Improved and expanded presentation. In estimate (2) now only a weaker error term than before is obtaine

On the Normality of Numbers to Different Bases

We prove independence of normality to different bases We show that the set of real numbers that are normal to some base is Sigma^0_4 complete in the Borel hierarchy of subsets of real numbers. This was an open problem, initiated by Alexander Kechris, and conjectured by Ditzen 20 years ago

Old and new results on normality

We present a partial survey on normal numbers, including Keane's contributions, and with recent developments in different directions.Comment: Published at http://dx.doi.org/10.1214/074921706000000248 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Matrices commuting with a given normal tropical matrix

Consider the space $M_n^{nor}$ of square normal matrices $X=(x_{ij})$ over $\mathbb{R}\cup\{-\infty\}$, i.e., $-\infty\le x_{ij}\le0$ and $x_{ii}=0$. Endow $M_n^{nor}$ with the tropical sum $\oplus$ and multiplication $\odot$. Fix a real matrix $A\in M_n^{nor}$ and consider the set $\Omega(A)$ of matrices in $M_n^{nor}$ which commute with $A$. We prove that $\Omega(A)$ is a finite union of alcoved polytopes; in particular, $\Omega(A)$ is a finite union of convex sets. The set $\Omega^A(A)$ of $X$ such that $A\odot X=X\odot A=A$ is also a finite union of alcoved polytopes. The same is true for the set $\Omega'(A)$ of $X$ such that $A\odot X=X\odot A=X$. A topology is given to $M_n^{nor}$. Then, the set $\Omega^{A}(A)$ is a neighborhood of the identity matrix $I$. If $A$ is strictly normal, then $\Omega'(A)$ is a neighborhood of the zero matrix. In one case, $\Omega(A)$ is a neighborhood of $A$. We give an upper bound for the dimension of $\Omega'(A)$. We explore the relationship between the polyhedral complexes $span A$, $span X$ and $span (AX)$, when $A$ and $X$ commute. Two matrices, denoted $\underline{A}$ and $\bar{A}$, arise from $A$, in connection with $\Omega(A)$. The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.Comment: Journal versio