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We show that there is a low T-upper bound for the class of K-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in the T-degrees below 0 for which there is a low T-upper bound.
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We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $tin {1,2}$.
We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings. Let $R$ be any ring equipped with an arbitrary additional first order structure, and $A$ a set of parameters. We show that whenever $H$ is an $A$-definable, finite index subgroup of $(R,+)$, then $H+RH$ contains an $A$-definable, two-sided ideal of finite index. As a corollary, we positively answer Question 3.9 of [Bohr compactifications of groups and rings, J. Gismatullin, G. Jagiella and K. Krupinski]: if $R$ is unital, then $(bar R,+)^{00}_A + bar R cdot (bar R,+)^{00}_A + bar R cdot (bar R,+)^{00}_A = bar R^{00}_A$, where $bar R succ R$ is a sufficiently saturated elementary extension of $R$, and $(bar R,+)^{00}_A$ [resp. $bar R^{00}_A$] is the smallest $A$-type-definable, bounded index additive subgroup [resp. ideal] of $bar R$. This implies that $bar R^{00}_A=bar R^{000}_A$, where $bar R^{000}_A$ is the smallest invariant over $A$, bounded index ideal of $bar R$. If $R$ is of finite characteristic (not necessarily unital), we get a sharper result: $(bar R,+)^{00}_A + bar R cdot (bar R,+)^{00}_A = bar R^{00}_A$. We obtain similar results for finitely generated (not necessarily unital) rings and for topological rings. The above results imply that the simplified descriptions of the definable (so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 3.5 of the aforementioned paper are valid for all unital rings. We analyze many examples, where we compute the number of steps needed to generate a group by $(bar R cup {1}) cdot (bar R,+)^{00}_A$ and study related aspects, showing optimality of some of our main results and answering some natural questions.
We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.
Suppose that a solution $widetilde{mathbf{x}}$ to an underdetermined linear system $mathbf{b} = mathbf{A} mathbf{x}$ is given. $widetilde{mathbf{x}}$ is approximately sparse meaning that it has a few large components compared to other small entries. However, the total number of nonzero components of $widetilde{mathbf{x}}$ is large enough to violate any condition for the uniqueness of the sparsest solution. On the other hand, if only the dominant components are considered, then it will satisfy the uniqueness conditions. One intuitively expects that $widetilde{mathbf{x}}$ should not be far from the true sparse solution $mathbf{x}_0$. We show that this intuition is the case by providing an upper bound on $| widetilde{mathbf{x}} - mathbf{x}_0|$ which is a function of the magnitudes of small components of $widetilde{mathbf{x}}$ but independent from $mathbf{x}_0$. This result is extended to the case that $mathbf{b}$ is perturbed by noise. Additionally, we generalize the upper bounds to the low-rank matrix recovery problem.
Let $q$ be a power of a prime $p$, let $k$ be a nontrivial divisor of $q-1$ and write $e=(q-1)/k$. We study upper bounds for cyclotomic numbers $(a,b)$ of order $e$ over the finite field $mathbb{F}_q$. A general result of our study is that $(a,b)leq 3$ for all $a,b in mathbb{Z}$ if $p> (sqrt{14})^{k/ord_k(p)}$. More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: $(0,0), (0,a), (a,0), (a,a)$ and $(a,b)$, where $a eq b$ and $a,b in {1,dots,e-1}$. The main idea we use is to transform equations over $mathbb{F}_q$ into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers.
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