Search

In this note we generalize a criterion within the concept of infinite dense lineability due to Calder\'on-Moreno, Gerlach-Mena and Prado-Bassas. We also introduce and explore some local notions of lineability.

We provide two general spaceability criteria, and we apply them to several specific topological vector spaces. As a consequence of our main results, we can extend and verify several classical results of this theory, as well as we can ensure the existence, except for zero, of large algebraic structures within, for example, spaces of Lebesgue measurable or continous functions, of several sequence spaces, of H\"older or Sobolev spaces, spaces of non-absolutely summing operators or even in the space of functions of bounded variation.

In this note we prove that, for $p>0$, $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ is $(\alpha,\mathfrak{c})$-spaceable if, and only if, $\alpha<\aleph_{0}$. Such a problem first appears in [V. F\'avaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46], where the authors get the $(1,\mathfrak{c})$-spaceability of $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ for $p>0$. The definitive answer to this problem continued to be sought by other authors, and some partial answers were obtained. The veracity of this result was expected, as a similar result is known for sequence spaces.

The investigation of regularity/summability properties of the coefficients of bilinear forms in sequence spaces was initiated by Littlewood in $1930$. Nowadays, this topic has important connections with other fields of Pure and Applied Mathematics as Complex Analysis, Quantum Information Theory, Theoretical Computer Science and Combinatorial Games. In this paper we explore a regularity technique to obtain optimal parameters for several results in this framework, extending/generalizing theorems of Osikiewicz and Tonge ($2001$), Albuquerque \textit{et al.} ($2016$), Aron \textit{et al.} ($2017$), Albuquerque and Rezende ($2018$), Paulino ($2020$), among others.

In $1977$, G. Bennett proved, by means of non-deterministic methods, an inequality which plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for $p_{1},p_{2} \in\lbrack1,\infty]$ and all positive integers $n_{1},n_{2}$, there exists a bilinear form $A_{n_{1},n_{2}}\colon\left( \mathbb{R}^{n_{1}},\left\Vert \cdot\right\Vert _{p_{1}}\right) \times\left( \mathbb{R}^{n_{2}},\left\Vert \cdot\right\Vert _{p_{2}}\right) \longrightarrow\mathbb{R}$ with coefficients $\pm1$ satisfying \[ \left\Vert A_{n_{1},n_{2}}\right\Vert \leq C_{p_{1},p_{2}}\max\left\{ n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max\left\{ \frac{1}{2}-\frac{1}{p_{2} },0\right\} },n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max\left\{ \frac{1}{2} -\frac{1}{p_{1}},0\right\} }\right\} \] for a certain constant $C_{p_{1},p_{2}}$ depending just on $p_{1},p_{2}$; moreover, the exponents of $n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach, we prove that $C_{p_{1},p_{2}}\leq\sqrt{8/5}$ whenever $p_{1},p_{2}\in\left[ 2,\infty\right] $ or $p_{1}=p_{2}=p\in\left[ 1,\infty\right] $. Our techniques are applied to provide new upper bounds for the constants of a combinatorial game, known as Gale--Berlekamp switching game or unbalancing lights problem. As a consequence, we improve estimates obtained by Brown and Spencer in $1971$ and by Carlson and Stolarski in $2004$., Comment: arXiv admin note: text overlap with arXiv:2006.12892

In [18] we have shown that, for $p_{1},p_{2}\in(2,\infty]$, the constants of Bennett's inequality on unimodular bilinear forms on $\ell_{p_{1}}^{n_{1} }\times\ell_{p_{2}}^{n_{2}}$ are asymptotically bounded by $1$. In the present paper we use a different approximation technique to investigate the remaining cases $p_{1},p_{2}\in\lbrack1,\infty].$ This new approach also provides a stronger asymptotic control, in the sense that the constants are "uniformly" asymptotically bounded by $1$, with no dependence on $p_{1},p_{2}.$, Comment: This paper is now incorporated to the new version of arXiv:2006.12892

We prove that several results of lineability/spaceability in the framework of sequence spaces are valid in a stricter sense.

In this note we introduce a new technique to answer an issue posed in [7] concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper., Comment: 7 pages

The Kahane--Salem--Zygmund inequality for multilinear forms in $\ell_{\infty}$ spaces claims that, for all positive integers $m,n_{1},...,n_{m}$, there exists an $m$-linear form $A\colon\ell_{\infty}^{n_{1}}\times\cdots\times \ell_{\infty}^{n_{m}}\longrightarrow\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) of the type \[ A(z^{(1)},...,z^{(m)})=\sum_{j_{1}=1}^{n_{1}}\cdots\sum_{j_{m}=1}^{n_{m}}\pm z_{j_{1}}^{\left( 1\right) }\cdots z_{j_{m}}^{\left( m\right) }\text{,} \] satisfying \[ \Vert A\Vert\leq C_{m}\max\left\{ n_{1}^{1/2},\ldots,n_{m}^{1/2}\right\} {\textstyle\prod\limits_{j=1}^{m}}n_{j}^{1/2}\text{,} \] for \[ C_{m}\leq\kappa\sqrt{m\log m}\sqrt{m!} \] and a certain $\kappa>0.$ Our main result shows that given any $\epsilon>0$ and any positive integer $m,$ there exists a positive integer $N$ such that \[ C_{m}<1+\epsilon\text{,} \] when we consider $n_{1},...,n_{m}>N$. In addition, while the original proof of the Kahane--Salem--Zygmund relies in highly non-deterministic arguments, our approach is constructive. We also provide the same asymptotic bound (which is shown to be optimal in some cases) for the constant of a related non-deterministic inequality proved by G. Bennett in 1977. Applications to Berlekamp's switching game are given., Comment: This version incorporates the results presented in arXiv:2104.10765 (the preprint arXiv:2104.10765 was withdrawn)

We investigate algebraic structures within sets of surjective and injective linear operators between sequence spaces, completing results of Aron et al.