---
title: 'Achievement Set: Theory & Applications'
url: https://www.emergentmind.com/topics/achievement-set
type: topic
---

# Achievement Set: Theory & Applications

Searching arXiv for recent and foundational uses of “achievement set” across mathematics, games, and applied ML/HCI.
Across the cited literature, the term **achievement set** denotes several non-equivalent objects. Its classical meaning is the set of all subsums of a convergent series, usually for an absolutely convergent real sequence; in later work it is extended to conditionally convergent series, finite-dimensional vector series, planar subsum sets, hyperspace operators, and several families of achievement-style games. In applied settings, the term is also used for finite sets of in-environment achievements, trophy lists, or realized-goal subsets, as in reinforcement learning, digital games, and self-tracking systems [2512.17285] [2312.08155] [2305.00508] [2205.15163] [1904.02813].

## 1. Classical subsum-set meaning

For an absolutely convergent series \(\sum x_n\), the classical achievement set is the set of all subseries sums,
\[
E(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\}\right\},
\]
equivalently
\[
E(x_n)=\Bigl\{y\in\mathbb R:\exists\,A\subset\mathbb N\text{ such that }y=\sum_{n\in A}x_n\Bigr\}.
\]
This is the basic definition used in the modern survey literature and in recent work on positive convergent series [2512.17285] [2512.17761]. For signed real series, the one-dimensional theory reduces to the nonnegative case via
\[
E(x_n)=E(|x_n|)+\sum_{n:\,x_n<0}x_n,
\]
so many topological questions can be studied under the assumption \(x_n\ge 0\) [2512.17285].

Kakeya’s classical results remain foundational in this setting. For an absolutely summable nontrivial real sequence, \(E(x_n)\) is compact and perfect; if \(|x_n|>\sum_{k=n+1}^\infty |x_k|\) for every \(n\), then \(E(x_n)\) is homeomorphic to the Cantor set; if \(|x_n|\le \sum_{k=n+1}^\infty |x_k|\) for all but finitely many \(n\), then \(E(x_n)\) is a finite union of closed intervals [1712.01706] [2512.17285]. A standard tail notation is
\[
r_n=\sum_{i=n+1}^\infty x_i,
\]
and much of the theory is governed by the comparison between \(x_n\) and \(r_n\) [2412.00042] [2512.17285].

The survey literature fixes the modern one-dimensional classification: for an absolutely convergent series, the achievement set is exactly one of four types—finite set, multi-interval set, Cantor set, or Cantorval [2512.17285]. In the positive nonincreasing case, the trichotomy interval / Cantor / Cantorval is often expressed in terms of the indices where \(x_n>r_n\) and \(x_n\le r_n\) [2512.17761].

## 2. Topological types, Kakeya conditions, and multigeometric regimes

A central invariant is the set of Kakeya indices
\[
K(x_n):=\{n\in\mathbb N:\ x_n>r_n\}.
\]
Classically, \(E(x_n)\) is a multi-interval set if and only if \(K(x_n)\) is finite, while \(E(x_n)\) is a Cantor set if \(K(x_n)\) has finite complement [2512.17285]. What happens when both \(K(x_n)\) and its complement are infinite is subtler. The Guthrie–Nymann example
\[
c_{2n-1}=\frac{3}{4^n},\qquad c_{2n}=\frac{2}{4^n}
\]
has achievement set a Cantorval and
\[
K(c_n)=2\mathbb N,
\]
whereas
\[
b_{2k}=b_{2k-1}=\frac1{4^k}
\]
has a Cantor-set achievement set with the same Kakeya pattern [2512.17285]. Recent work strengthens this non-classifiability: for every infinite \(K\subset\mathbb N\) with infinite complement, one can realize \(K(a_n)=K\) both with \(E(a_n)\) a Cantor set and with \(E(a_n)\) a Cantorval [2512.17285] [2512.17761].

Within structured families, sharper parameter criteria are available. For generalized multigeometric series
\[
k_1f(x)+\dots+k_mf(x)+k_1f(x^2)+\dots+k_mf(x^2)+\dots
\]
with \(f\) locally increasing and power bounded at \(0\), the paper “The achievement set of generalized multigeometric sequences” gives explicit threshold constants \(d_I,d_{IM},d_{NI},d_{CI},d_C\) separating interval, non-interval, interval-containing, and Cantor regimes [2309.11388]. In particular, for sufficiently large \(x\), \(E(w_n(x))\) is a compact interval; for sufficiently small \(x\), it is homeomorphic to the Cantor set; and under an arithmetic richness condition on block-subsums, there is an intermediate parameter region where \(E(w_n(x))\) is a Cantorval [2309.11388].

A distinct recent development is the generalized Ferens framework. If a convergent GF series satisfies the explicit block inequalities labeled (GF1) and (GF2), then its achievement set is a Cantorval; this yields new achievable Cantorvals outside the multigeometric class [2309.01589]. The same paper proves two strong addition theorems: there exists an achievable Cantorval whose \(k\)-fold algebraic sum remains a Cantorval for every finite \(k\), and for any \(m,p\in(\mathbb N\setminus\{1\})\cup\{\infty\}\) with \(p\ge m\), there exists an achievable Cantor set \(C\) such that \(C^k\) is a Cantor set for \(k<m\), a Cantorval for \(m\le k<p\), and an interval for \(k\ge p\) [2309.01589].

The 2025 paper “On a new condition implying that an achievement set is a Cantorval and its applications” introduces the **Star Procedure**, a recursive sufficient criterion formulated in terms of overlap lengths \(\delta_i=r_i-a_i\) and auxiliary quantities \(M_i^n\). If the inequalities \((*)\), \((**)\), and \((*')\) can be continued indefinitely, then \(E(a_n)\) is a Cantorval [2512.17761]. This criterion recovers classical examples such as \(E(3,2;1/4)\), yields new multigeometric Cantorvals, and proves that Kakeya conditions alone cannot classify the mixed regime more finely than the original Kakeya theorems [2512.17761].

## 3. Conditionally convergent and finite-dimensional extensions

For conditionally convergent series, the real-line picture is simpler than the absolute theory: for a conditionally convergent real series,
\[
A(x_n)=\mathbb R,
\]
and more generally if a real series is potentially conditionally convergent, then its achievement set is \(\mathbb R\) [1712.01706] [1604.07575]. In finite-dimensional spaces, however, the structure becomes much richer.

The paper “Subsums of conditionally convergent series in finite dimensional spaces” studies
\[
A(x_n)=\left\{\sum_{n=1}^\infty \varepsilon_n x_n:\ \varepsilon_n\in\{0,1\},\ \sum \varepsilon_n x_n \text{ converges}\right\}
\]
and the absolute subsum set \(A_{\mathrm{abs}}\). Its main theorem states that for pairwise distinct \(\alpha_1,\dots,\alpha_d\in(0,1]\) and nonzero \(a_1,\dots,a_d\),
\[
A_{\mathrm{abs}\bigl(a_1(-1)^{n+1}n^{-\alpha_1},\dots,a_d(-1)^{n+1}n^{-\alpha_d}\bigr)=\mathbb R^d
\]
if and only if the exponents are pairwise distinct; consequently the full achievement set is also \(\mathbb R^d\) [1802.10535]. A highlighted special case is
\[
A\left(\frac{(-1)^n}{n},\frac{(-1)^n}{\sqrt n}\right)=\mathbb R^2,
\]
which had previously been open [1802.10535].

The geometry of Levy vectors gives a complementary description in \(\mathbb R^2\). If a conditionally convergent planar series has more than two Levy vectors, then
\[
A_{\mathrm{abs}}(x_n)=A(x_n)=\mathbb R^2.
\]
If it has exactly two Levy vectors, several behaviors are possible: \(A(x_n)\) can still equal \(\mathbb R^2\); one can have \(A_{\mathrm{abs}}(x_n)\subsetneq A(x_n)\); and there are examples with \(SR(x_n)=\mathbb R^2\) but achievement set extremely small, even with each vertical section containing at most one point [1705.06472]. This phenomenon is one reason the survey literature treats the two-Levy-vector case as especially delicate [2512.17285].

More generally, for a conditionally convergent series \((x_n)\subset\mathbb R^k\) with full sum range \(SR(x_n)=\mathbb R^k\) and an absolutely convergent series \((y_n)\subset\mathbb R^m\), one has the product formula
\[
A(x_n,y_n)=\mathbb R^k\times A(y_n).
\]
This yields examples such as \(\mathbb R\times C\), graph-like sets, nonclosed sets, and proper open strips in higher-dimensional conditional theory [1604.07575].

## 4. Planar achievement sets, hyperspaces, and the spectre

For an absolutely convergent planar series,
\[
E(x_n,y_n)=\left\{\sum_{n=1}^{\infty}\varepsilon_n (x_n,y_n):\varepsilon_n\in\{0,1\}\right\}\subset\mathbb R^2,
\]
the basic structural facts are that
\[
E(x_n,y_n)\subset E(x_n)\times E(y_n),\qquad pr_x E(x_n,y_n)=E(x_n),\qquad pr_y E(x_n,y_n)=E(y_n),
\]
and \(E(x_n,y_n)\) is compact and centrally symmetric [2312.08155]. Unlike the one-dimensional case, these sets need not be products, and the paper “Achievement sets of series in \(\mathbb R^2\)” shows that planar sections can realize far more complicated one-dimensional objects.

The main cut theorem states that for every finite \(P\subset\mathbb R\) with \(0\in P\) and every absolutely convergent real sequence \(a=(a_n)\) with \(|a_n|\ge |a_{n+1}|>0\), there exists a planar series \((x_n,y_n)\) such that
\[
E(x_n,y_n)_0=S(P,a),
\]
where \(S(P,a)\) is the corresponding set of \(P\)-sums [2312.08155]. A consequence is that an \(L\)-Cantorval can occur as a horizontal section of a planar achievement set, so the planar theory is not exhausted by product types such as \(I\times I\), \(I\times M\), \(C\times M\), or \(M\times M\) [2312.08155].

A second major innovation is the **spectre**. For an Abelian group \((X,+)\) and \(A\subset X\),
\[
S(A):=\{x\in X:\forall_{y\in A}\ (y+x\in A \text{ or } y-x\in A)\}.
\]
On \(\mathbb R\), \(S(A)\cap[0,\infty)\) coincides with the center of distances, but the spectre is vector-valued and well adapted to \(\mathbb R^2\) [2312.08155]. For every absolutely convergent series, each term belongs to the spectre of its achievement set:
\[
x_n\in S(E(x_n)).
\]
In the planar case, if \(A=[0,1]^2\), then
\[
S(A)= \left(\{0\}\times\left[-\frac12,\frac12\right]\right)\cup \left(\left[-\frac12,\frac12\right]\times\{0\}\right),
\]
while for the Sierpiński carpet one has
\[
S(SC)=\{(0,0)\},
\]
which implies that the Sierpiński carpet is not the achievement set of any series [2312.08155].

The hyperspace paper “Spectre operator, achievement sets and sets of \(P\)-sums in a hyperspace of compact sets” places these objects inside \(K([0,1])\) with the Hausdorff metric. It proves that the family \(\mathcal A\) of achievement sets in \(K([0,1])\) is closed and hence nowhere dense, whereas the family \(\mathcal P\) of sets of \(P\)-sums is not closed [2512.11803]. The same paper establishes monotonicity properties of the spectre along finite partial-sum sets \(F_n\) and tail achievement sets \(E_n\), and proves planar analogues of the first two gap lemmas while showing that the one-dimensional third-gap phenomenon fails in \(\mathbb R^2\) [2512.11803].

## 5. Achievement sets in games and combinatorial structures

In several game-theoretic literatures, “achievement” refers not to subsums but to the family of positions satisfying a winning target. The resulting “achievement set” is therefore positional rather than analytic.

In convex-geometry game theory, the game \(GEN(S,W)\) is played on a finite convex geometry \((S,\mathcal K)\) with closure operator
\[
\tau(A)=\bigcap\{K\in\mathcal K:A\subseteq K\}.
\]
The relevant achievement family is
\[
\{P\subseteq S: W\subseteq \tau(P)\},
\]
that is, the generating positions whose convex closure already contains the winning set \(W\) [2010.11319]. The paper “Impartial Achievement Games on Convex Geometries” develops a structure theory based on maximally non-generating sets, intersection subsets, and parity-sensitive structure classes, and derives explicit nim-value formulas for extreme-point targets, vertex geometries of trees, and affine geometries in one dimension [2010.11319].

In positional game theory, the paper “A unified convention for achievement positional games” formalizes an achievement positional game as
\[
G=(V,E_L,E_R),
\]
where \(E_L\) and \(E_R\) are the blue and red winning hyperedges of the two players [2503.18163]. Here an “achievement set” is naturally a player-specific winning hyperedge, or more broadly the player’s winning family \(E_L\) or \(E_R\). The paper proves that many Maker–Maker principles extend to this asymmetric setting and classifies the complexity of deciding whether Left wins as first player: the problem is in \({\sf P}\) for \(p,q\le 2\), NP-hard for \(p\ge 3,q=2\), coNP-complete for \(p=2,q\ge 3\), and PSPACE-complete for \(p,q\ge 3\) [2503.18163].

Other combinatorial achievement games make the same shift from subsums to target families. In the weak polyomino set \((1,2)\)-achievement game, the maker tries to occupy cells congruent to one of a target set of polyominoes, and the paper proves that every winning team of cardinality \(<5\) is simpler than a specific unbounded “super winner” \(\mathcal W\) [1010.0424]. In the general position achievement game on graphs, the evolving chosen-vertex set must remain a general position set, and the last legal move wins [2111.07425]. These uses are terminologically consistent with the idea of “achieving” membership in a target family, but they are structurally distinct from the subsum-set tradition.

## 6. Applied reinterpretations in tracking, games, and reinforcement learning

Applied literatures use the phrase in still different ways. In self-tracking, the MyFitnessPal study defines the “achievement set” as the subset of weight-loss goals that are eventually realized: for a goal weight \(w_g\), a goal is achieved if at some later time the user logs a weight less than or equal to \(w_g\) [1904.02813]. On that operational definition, the study of 1,413,431 users reports that only 18.2% of weight-loss goals are achieved, and that a Random Forest model using the first 7 days of behavior predicts eventual achievement with 79% ROC AUC [1904.02813]. This usage is not a set of subsums but a realized-goal subset of observed user-goal episodes.

In reinforcement learning, an **achievement-based environment** is formalized as a Markov Decision Process with Achievements
\[
(\mathcal S,\mathcal A,T,\Gamma,G),
\]
where \(\Gamma\) is the finite internal set of achievements and \(G\) is the achievement completion function [2305.00508]. SEA first learns an embedding of reward-triggering transitions with the determinant loss
\[
L_\text{achv}(\theta):=E_\tau\left[-\det\left(\exp\left(-k\overline{D}_\tau\right)\right)\right],
\]
then clusters the known achievements and recovers a dependency DAG using
\[
G_{ij}:=1\{Before_{ij}/Happen_j>1-\epsilon\},
\]
after which a graph-based controller uses the recovered structure for exploration [2305.00508]. In this setting, the achievement set is a latent but finite set of reusable events rather than a topological subset of \(\mathbb R\).

In digital-game analytics, the PlayStation study treats a game’s trophy list as its achievement set. At platform scale, the dataset contains 13,792 games and 377,938 trophies, with trophy-point structure Bronze \(=15\), Silver \(=30\), Gold \(=90\), Platinum \(=180\) [2205.15163]. The paper identifies strong platform conventions: trophy-score totals cluster around 300 and 1,200, “L” games are defined by total score \(<750\), “H” games by total score \(\ge 750\), and larger games typically have exactly one Platinum trophy [2205.15163]. Here the achievement set is a designed finite catalog rather than a generated mathematical set.

Taken together, these applied uses suggest a broader semantic pattern: “achievement set” often denotes the finite family of achievements available in a system, or the subset of them realized by an agent. A plausible implication is that the mathematical and applied literatures share the vocabulary of attainable states while differing sharply in ontology: subsums in analysis, winning positions in games, and labeled goals or events in HCI and RL.

Source: https://www.emergentmind.com/topics/achievement-set