Theorem For Subsets Research Articles

Bounding Radon Numbers via Betti Numbers

Abstract We prove general topological Radon-type theorems for sets in $\mathbb R^{d}$ or on a surface. Combined with a recent result of Holmsen and Lee, we also obtain fractional Helly theorem, and consequently the existence of weak $\varepsilon $-nets as well as a $(p,q)$-theorem for those sets. More precisely, given a family ${\mathcal{F}}$ of subsets of ${\mathbb{R}}^{d}$, we will measure the homological complexity of ${\mathcal{F}}$ by the supremum of the first $\lceil d/2\rceil $ reduced Betti numbers of $\bigcap{\mathcal{G}}$ over all nonempty ${\mathcal{G}} \subseteq{\mathcal{F}}$. We show that if ${\mathcal{F}}$ has homological complexity at most $b$, the Radon number of ${\mathcal{F}}$ is bounded in terms of $b$ and $d$. In case that ${\mathcal{F}}$ lives on a surface and the number of connected components of $\bigcap \mathcal G$ is at most $b$ for any $\mathcal G\subseteq \mathcal F$, then the Radon number of ${\mathcal{F}}$ is bounded by a function depending only on $b$ and the surface itself. For surfaces, if we moreover assume the sets in ${\mathcal{F}}$ are open, we show that the fractional Helly number of $\mathcal F$ is linear in $b$. The improvement is based on a recent result of the author and Kalai. Specifically, for $b=1$ we get that the fractional Helly number is at most three, which is optimal. This case further leads to solving a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.

Remarks on “some fixed point theorems for s-convex subsets in p-normed spaces”

Remarks on “some fixed point theorems for s-convex subsets in p-normed spaces”

A type of minimal and maximal point theorem in locally convex product cones

Using the neighbourhoods in upper relative topologies, we introduce a quasi-metric on cones which leads to a version of Cantors's intersection theorem for upper and lower relative v-topologies. Then we prove a type of minimal and maximal point theorem for subsets of product cone topologies and obtain the corresponding Ekeland's variational principle (EVP) results for locally convex cone-valued functions.

Approximate subgroups with bounded VC-dimension

We combine the fundamental results of Breuillard, Green, and Tao (Publ Math Inst Hautes Études Sci 116:115–221, 2012) on the structure of approximate groups, together with “tame” arithmetic regularity methods based on work of the authors and Terry (J Eur Math Soc (JEMS) 24(2):583–621, 2022), to give a structure theorem for finite subsets A of arbitrary groups G where A has “small tripling” and bounded VC-dimension: Roughly speaking, up to a small error, A will be a union of a bounded number of translates of a coset nilprogression of bounded rank and step (see Theorem 2.1). We also prove a stronger result in the setting of bounded exponent (see Theorem 2.2). Our results extend recent work of Martin-Pizarro, Palacín, and Wolf (Selecta Math (N.S.) 27(4):Paper No. 53, 19,2021) on finite stable sets of small tripling.

A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space H. We prove a version of Azzam and Schul’s d-dimensional Analyst’s Travelling Salesman Theorem in this setting by showing for any lower d-regular set \(E \subseteq H\) that $$\begin{aligned} \textrm{diam}(E)^d + \beta ^d(E) \sim \mathscr {H}^d(E) + \text {Error}, \end{aligned}$$where \(\beta ^d(E)\) give a measure of the curvature of E and the error term is related to the theory of uniform rectifiability (a quantitative version of rectifiability introduced by David and Semmes). To do this, we show how to modify the Reifenberg Parametrization Theorem of David and Toro so that it holds in Hilbert space. As a corollary, we show that a set \(E \subseteq H\) is uniformly rectifiable if and only if it satisfies the so-called Bilateral Weak Geometric Lemma, meaning that E is bi-laterally well approximated by planes at most scales and locations.

A Szemerédi-type theorem for subsets of the unit cube

We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit bound for the length of the aforementioned interval. The proof makes the bound depend on the currently available bounds in Szemer\'{e}di's theorem on the integers, which are used as a black box. A key ingredient of the proof are power-type cancellation estimates for operators resembling the multilinear Hilbert transforms. As a byproduct of the approach we obtain a quantitative improvement of the corresponding (previously known) result for side lengths of $n$-dimensional cubes with vertices lying in a positive measure subset of $([0,1]^2)^n$.

Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces

<abstract> <p>In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.</p> </abstract>

Small and Large Inductive Dimensions of Intuitionistic Fuzzy Topological Spaces

The main purpose of this paper is to present some fundamental properties of small and large inductive dimensions in intuitionistic fuzzy topological spaces. Our results can be regarded as a study of their properties such as proves subset theorems, zero dimensionality and topological property of an intuitionistic fuzzy small inductive dimension. Furthermore, we introduce a large inductive dimension of intuitionistic fuzzy bi-compact and normal spaces.

Embedding properties of sets with finite box-counting dimension

In this paper we study the regularity of embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces. In 1999, Hunt and Kaloshin (1999 Nonlinearity 12 1263–75) introduced the thickness exponent and proved an embedding theorem for subsets of Hilbert spaces with finite box-counting dimension. In 2009, Robinson (2009 Nonlinearity 22 711–28) defined the dual thickness and extended the result to subsets of Banach spaces. Here we prove a similar result for subsets of Banach spaces, using the thickness rather than the dual thickness. We also study the relation between the box-counting dimension and these two thickness exponents for some particular subsets of .

Algebraic stratified general position and transversality

The method of Whitney interpolation is used to construct, for any real or complex projective algebraic variety, a stratified submersive family of self-maps that yields stratified general position and transversality theorems for semialgebraic subsets.

Some fixed point theorems for s-convex subsets in p-normed spaces based on measures of noncompactness

The Kurotowskii and Hausdorff measures of noncompactness in p-normed spaces are studied, where $$0<p\le 1$$ . By metric retraction and calculation of measures of noncompactness, some fixed point theorems for s-convex subsets are proved, with respect to single-valued and set-valued condensing operators, where $$0<s\le p$$ . The results herein extend and generalize some of the well-known fixed point theorems, such as the types of Darbo, Sadovski, Leray-Schauder, and Martelli.

On a theorem of Deshouillers and Freiman

The “structure theory of set addition” has been well studied in the last fifty years, after the works of Freiman. In [1] Deshouillers and Freiman established a structure theorem for subsets of Z∕nZ with small doubling constant. In this article we provide a simpler proof of the structure theorem of [1], and on our way we also obtain some improvements.

The Closed Subset Theorem for Inverse Limits with Upper Semicontinuous Bonding Functions

We give several characterizations of inverse limits of compact metric spaces with upper semicontinuous set-valued bonding functions having the property that any closed subset of the inverse limit is the inverse limit of its projections. This solves a problem stated by Ingram.

A Roth‐type theorem for dense subsets of Rd

Let $1 < p < \infty$, $p\neq 2$. We prove that if $d\geq d_p$ is sufficiently large, and $A\subs\R^d$ is a measurable set of positive upper density then there exists $\la_0=\la_0(A)$ such for all $\la\geq\la_0$ there are $x,y\in\R^d$ such that $\{x,x+y,x+2y\}\subs A$ and $|y|_p=\la$, where $||y||_p=(\sum_i |y_i|^p)^{1/p}$ is the $l^p(\mathbb R^d)$-norm of a point $y=(y_1,\ldots,y_d)\in\R^d$. This means that dense subsets of $\R^d$ contain 3-term progressions of all sufficiently large gaps when the gap size is measured in the $l^p$-metric. This statement is known to be false in the Euclidean $l^2$-metric as well as in the $l^1$ and $\ell^{\infty}$-metrics. One of the goals of this note is to understand this phenomenon. A distinctive feature of the proof is the use of multilinear singular integral operators, widely studied in classical time-frequency analysis, in the estimation of forms counting configurations.

ON LOCAL HUDETZ g-ENTROPY

In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.

Szemerédi-type theorems for subsets of locally compact abelian groups of positive upper Banach density

By using ergodic theoretic techniques following Hillel Furstenberg, we prove that measurable subsets of a locally compact abelian group of positive upper density contain Szemeredi-wise configurations defined by an arbitrary compact subset of the group.

Sum-free subsets of finite abelian groups of type III

A finite abelian group G of cardinality n is said to be of type III if every prime divisor of n is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group G of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group G. We supplement this result with a theorem on the structure of sum-free subsets of cardinality “close” to the largest possible in a type III abelian group G. We then give two applications of these results. Our first application allows us to write down a formula for the number of orbits under the natural action of Aut(G) on the set of sum-free subsets of G of the largest cardinality when G is of the form (Z/mZ)r, with all prime divisors of m congruent to 1 modulo 3, thereby extending a result of Rhemtulla and Street. Our second application provides an upper bound for the number of sum-free subsets of G. For finite abelian groups G of type III and with a given exponent this bound is substantially better than that implied by the bound for the number of sum-free subsets in an arbitrary finite abelian group, due to Green and Ruzsa.

Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers

We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p&gt;2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.

QUASIRANDOM GROUP ACTIONS

Let $G$ be a finite group acting transitively on a set $\unicode[STIX]{x1D6FA}$. We study what it means for this action to be quasirandom, thereby generalizing Gowers’ study of quasirandomness in groups. We connect this notion of quasirandomness to an upper bound for the convolution of functions associated with the action of $G$ on $\unicode[STIX]{x1D6FA}$. This convolution bound allows us to give sufficient conditions such that sets $S\subseteq G$ and $\unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D6E5}_{2}\subseteq \unicode[STIX]{x1D6FA}$ contain elements $s\in S,\unicode[STIX]{x1D714}_{1}\in \unicode[STIX]{x1D6E5}_{1},\unicode[STIX]{x1D714}_{2}\in \unicode[STIX]{x1D6E5}_{2}$ such that $s(\unicode[STIX]{x1D714}_{1})=\unicode[STIX]{x1D714}_{2}$. Other consequences include an analogue of ‘the Gowers trick’ of Nikolov and Pyber for general group actions, a sum-product type theorem for large subsets of a finite field, as well as applications to expanders and to the study of the diameter and width of a finite simple group.

A cross‐intersection theorem for subsets of a set

A set of sets is called a family. Two families and are said to be cross-intersecting if each set in intersects each set in . For any two integers and with , let denote the family of all subsets of that have at most elements. We show that if is a subfamily of , is a subfamily of , and and are cross‐intersecting, then and equality holds if and .