Rough Weighted Ideal Limit Set

• Rough weighted ideal limit set is a generalization of classical convergence, integrating strictly positive weights, a roughness parameter, and admissible ideals on natural numbers.
• It refines convergence by allowing rough approximations up to prescribed degrees and filtering negligible index sets to capture broader convergence behaviors.
• The framework guarantees closed, convex, and bounded limit sets, extending applications from normed spaces to locally solid Riesz spaces.

The rough weighted ideal limit set is a set-valued generalization of limit concepts for sequences in normed spaces and locally solid Riesz spaces, incorporating both roughness (via a real parameter or neighborhood), weights (strictly positive sequences), and summability via admissible ideals on the natural numbers. This framework extends traditional convergence, cluster point characterizations, and summability by allowing "rough" approximation up to prescribed degrees, and by using ideals to filter out negligible index sets, thus refining and encompassing broader convergence behaviors.

1. Formal Definition and Fundamental Properties

Given a normed space $(X, \|\cdot\|)$, a sequence of strictly positive real weights $\{\omega_t\}_{t\in\mathbb{N}}$ (i.e., $\omega_t>\beta>0$ for all $t$), an admissible ideal $\mathcal{I}\subset\mathcal{P}(\mathbb{N})$ (containing finite sets), and a degree of roughness $r\ge 0$, the rough weighted ideal limit set of a sequence $(x_t)$ is defined as

$$\Lambda_{w}^{\mathcal{I},r}(x_t) = \{ x_* \in X : x_t \xrightarrow[r]{(\omega_t, \mathcal{I})} x_* \}$$

where $x_t \xrightarrow[r]{(\omega_t, \mathcal{I})} x_*$ if for every $\varepsilon>0$,

$$\left\{ t \in \mathbb{N} : \omega_t \|x_t - x_*\| > r+\varepsilon \right\} \in \mathcal{I}.$$

This set recovers classical rough limit sets in the case $\omega_t \equiv 1$ and $\mathcal{I} = \mathrm{Fin}$, and ideal limit sets without roughness for $r = 0$ (Aziz et al., 21 Dec 2025).

The set $\Lambda_{w}^{\mathcal{I},r}(x_t)$ is always closed, convex, and bounded (Lemma 2.9 in (Aziz et al., 21 Dec 2025)).

2. Equivalent Characterizations

For analytic $P$-ideals, $\mathcal{I}$, the rough weighted ideal limit set admits a subsequence formulation: $x_* \in \Lambda_{w}^{\mathcal{I},r}(x_t)$ if and only if there exists $A \subset \mathbb{N}$ with $\mathbb{N}\setminus A \in \mathcal{I}$ and

$\limsup_{t\in A} \omega_t \|x_t - x_*\| \le r$

(Lemma 2.2 in (Aziz et al., 21 Dec 2025)). This mirrors the classical subsequence characterization of ideal convergence, where the conditioning on large-index sets is governed by $\mathcal{I}$.

3. Minimal Convergent Degree and Non-emptiness

The minimal roughness degree for which the limit set is nonempty is

$$\tilde{r}(x) := \inf \{ r \ge 0 : \Lambda_{w}^{\mathcal{I},r}(x_t) \neq \varnothing \}.$$

Monotonicity holds: if $r_1 < r_2$, then $$\Lambda_{w}^{\mathcal{I},r_1}(x_t) \subseteq \Lambda_{w}^{\mathcal{I},r_2}(x_t)$$. The condition

$$\Lambda_{w}^{\mathcal{I},r}(x_t) = \varnothing \quad (r < \tilde r(x)), \qquad \Lambda_{w}^{\mathcal{I},r}(x_t) \neq \varnothing \quad (r > \tilde r(x))$$

is both necessary and sufficient.

If the weights are $\mathcal{I}$-bounded, the limit set even has nonempty interior for $r > \tilde r(x)$ (Proposition 3.5 in (Aziz et al., 21 Dec 2025)). For reflexive spaces, $\Lambda_{w}^{\mathcal{I},\tilde r(x)}(x_t)$ is always nonempty (Theorem 3.7), while in non-reflexive spaces this can fail for $r = \tilde r(x)$ (Example 3.8).

4. Borel Regularity and Set Structure

For $\mathcal{I} = \mathcal{I}_\varphi$ an analytic $P$-ideal, $\Lambda_{w}^{\mathcal{I},r}(x_t)$ is an $F_{\sigma\delta}$, and hence Borel, subset of $X$ (Proposition 2.3 in (Aziz et al., 21 Dec 2025)). The construction uses sets

$$U_{t,k} = \{ x \in X : \omega_t \|x_t - x\| > r + \tfrac{1}{k} \},$$

and writes

$$\Lambda_{w}^{\mathcal{I},r}(x_t) = \bigcap_{k=1}^\infty \{ x : \{ t : x \in U_{t,k} \} \in \mathcal{I}_\varphi \}$$

where membership in $\mathcal{I}_\varphi$ can be expressed as a countable $F_\sigma$, yielding the $F_{\sigma\delta}$ regularity.

5. Illustrative Examples and Noncompactness

• In $X = \ell^\infty$, with $x_t = e_t$ (the unit vectors) and $\omega_t \equiv \beta > 0$, for $r = \beta$, $\Lambda_{w}^{\mathcal{I}, r}(e_t)$ contains all finitely many $e_t$. This set is closed and bounded but not compact (Example 2.2 in (Aziz et al., 21 Dec 2025)).
• In $X = C[0,1]$ with $\mathcal{I} = \mathcal{I}_\delta$, one can construct $(x_t)$ with $\tilde r(x) = 0$ but $\Lambda_{w}^{\mathcal{I}, 0}(x_t) = \varnothing$ (Example 3.8), establishing that for non-reflexive spaces, limit set non-emptiness at $r = \tilde r(x)$ does not generally hold.

6. Relation to Cluster Sets and Maximal Ideals

The rough weighted ideal cluster set is

$$\Gamma_{w}^{\mathcal{I}, r}(x_t) = \{ \gamma \in X : \forall \varepsilon > 0, \{ t : \omega_t \| x_t - \gamma \| < r + \varepsilon \} \notin \mathcal{I} \}.$$

One always has

$$\Lambda_{w}^{\mathcal{I}, r}(x_t) \subseteq \Gamma_{w}^{\mathcal{I}, r}(x_t).$$

Generally, this inclusion is strict. The limit set $\Lambda_{w}^{\mathcal{I}, r}(x_t)$ is always closed, bounded, and convex, while the cluster set $\Gamma_{w}^{\mathcal{I}, r}(x_t)$ need not be closed (Example 4.2, (Aziz et al., 21 Dec 2025)), though it is closed if $\{\omega_t\}$ is $\mathcal{I}$-bounded or if $\mathcal{I}$ is maximal (Propositions 4.3, 4.4). For maximal admissible ideals, the limit and cluster sets coincide exactly: $$\Lambda_{w}^{\mathcal{I}, r}(x_t) = \Gamma_{w}^{\mathcal{I}, r}(x_t)$$ (Theorem 4.5).

7. Extensions to Locally Solid Riesz Spaces

The concept generalizes to locally solid Riesz spaces $(L, \tau)$, where roughness is given by a solid neighborhood $V$ of zero rather than a real parameter, and weighted ideal convergence is formulated relative to $\mathcal{I}_\tau$ and a topology-adapted definition (Ghosal et al., 2021). Main features include:

• Convexity of the limit set when $V$ is convex (Theorem 2.1).
• Uniqueness of the limit set under non-$\mathcal{I}$-bounded weights when $V$ is $\tau$-bounded (Theorem 2.2).
• Closedness, boundedness, and the algebraic structure of cluster points may fail without further hypotheses on weights or the space topology (Theorems 2.3–2.8).
• In this setting, classical statements about closure of cluster sets can fail (Example 9), and the interplay between the structure of $V$, the weights, and the topology yields new phenomena that extend and refine results from normed and metric settings (Ghosal et al., 2021).

8. Hierarchy and Summary Table

The following relationships summarize the hierarchy and major structural properties present in the literature:

Set	Closed	Convex	Coincides for Maximal $\mathcal{I}$
$\Lambda_{w}^{\mathcal{I}, r}(x_t)$	Yes	Yes	Yes
$\Gamma_{w}^{\mathcal{I}, r}(x_t)$	Not always	Not always	Yes

When both weight and ideal conditions are met (e.g., $\mathcal{I}$ maximal or weights $\mathcal{I}$-bounded), strong regularity properties for both sets can be ensured (Aziz et al., 21 Dec 2025).

---

These developments collectively provide a robust framework for rough convergence and clustering in modern analysis, extending the landscape of ideal convergence, weighted summability, and rough approximation across normed spaces, metric spaces, and vector lattices (Aziz et al., 21 Dec 2025, Ghosal et al., 2021).