A characterization of uniqueness of purely atomic finite measures with central Cantor set range

Abstract: We study purely atomic measures whose range is a central Cantor set and characterize those central Cantor sets that are the range of exactly one such measure. Next, we extend the characterization to the case of symmetric Cantor sets using a different method of proof. Finally, we make a few initial observations and remarks on recovering a measure whose range is a Cantorval.

• The paper proves that a central Cantor set is uniquely achievable exactly when its canonical sequence has no adjacent doubling relation, x_n ≠ 2x_{n+1}.
• The paper characterizes symmetric Cantor set uniqueness through irreducibility and multiplicities bounded by two, using dominating gaps and pseudointerval geometry rather than centers of distances.
• The paper establishes a sufficient alternating-gap condition for uniquely achievable Cantorvals, expands known examples, and identifies the general converse as an open problem.

This paper by Nowakowski and Prus-Wiśniowski addresses the problem of recovering a purely atomic finite measure from its range, equivalently, determining when the achievement set $E(x_n)$ of a summable, positive, non-increasing sequence $(x_n)$ is uniquely achievable, i.e., admits no other monotone representation. By the Guthrie–Nymann classification, such a range is either a Cantor set or a Cantorval; the authors give a complete characterization for central Cantor sets, extend it to general symmetric Cantor sets via a different method, and provide initial results for Cantorvals.

Background and setting

The range of a finite atomic measure on $\mathbb{N}$ with atom masses arranged non-increasingly coincides with the achievement set

$$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$

The standard toolkit includes the decomposition $E=E_1^k+E_k$ into $k$-initial subsums and tail achievements, the iterates $I_k=E_1^k+[0,r_k]$ (with $E=\bigcap_k I_k$), and the gap hierarchy: $\mathcal{G}_k$, the components of $I_{k-1}\setminus I_k$, is nonempty exactly when $(x_n)$0, in which case the First Gap Lemma makes $(x_n)$1 a principal gap. A gap is dominating if all gaps to its left are shorter, and the Third Gap Lemma states that every dominating gap is principal. Crucially, being a gap of order $(x_n)$2 depends on the chosen representation, whereas domination does not — a fact the authors exploit systematically.

The relevant prior work is that of Bartoszewicz, Głąb and Marchwicki [(2608.14395)'s cited BGM18], who gave sufficient conditions for unique achievability of Cantor sets, including fast convergence ($(x_n)$3 for all $(x_n)$4), answering a question of Banakh. The present paper sharpens these results to exact characterizations.

Central Cantor sets: characterization via centers of distances

Central Cantor sets correspond bijectively to fast convergent sequences: each central Cantor set with left endpoint 0 has a unique fast convergent representation $(x_n)$5 built from right endpoints of removed intervals, with remainders equal to left endpoints. Not every such set is uniquely achievable, however.

The main result (Theorem 4) is a clean iff criterion:

A central Cantor set $(x_n)$6 with its unique fast convergent representation $(x_n)$7 is uniquely achievable if and only if $(x_n)$8 for all $(x_n)$9.

The proof splits on whether the center of distances $\mathbb{N}$0 is minimal. The center of distances, introduced by Bielas–Plewik–Walczyńska as a metric invariant, satisfies $\mathbb{N}$1 always, with equality possible (e.g., geometric sequences with ratio below $\mathbb{N}$2). Banakiewicz's theorem characterizes non-minimality: it occurs exactly when one of three specific multiplicative patterns holds among consecutive terms, all of which force some $\mathbb{N}$3. Hence, if $\mathbb{N}$4 is not minimal, uniqueness fails. If $\mathbb{N}$5 is minimal but $\mathbb{N}$6 has another representation $\mathbb{N}$7, then every value of $\mathbb{N}$8 must lie among the $\mathbb{N}$9's; a counting argument shows some value $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$0 must appear at least three times in $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$1, whence $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$2 by the multiplicities lemma, forcing $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$3 under minimality — again violating the criterion. Conversely, if $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$4, splitting that term into two equal halves yields a distinct monotone representation.

An immediate consequence: within the central Cantor family, failure of unique achievability is entirely detected by a single local doubling relation. This also yields a corollary outside the stated theorem: the Sierpiński carpet computation $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$5, derived from a compact path-connectedness proposition, answers a question of Kula and illustrates that the center-of-distances invariant loses diagnostic power in dimension two — the authors note this favors the "spectre" concept instead.

Symmetric Cantor sets: irreducibility and bounded multiplicity

For symmetric Cantor sets $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$6, represented by semi-fast convergent sequences $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$7 with multiplicities, the authors define a value $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$8 to be reducible if $$E(x_n)=\Bigl\{\sum_{n\in A}x_n:\ A\subset\mathbb{N}\Bigr\}.$$9, and prove:

A symmetric Cantor set $E=E_1^k+E_k$0 is uniquely achievable if and only if $E=E_1^k+E_k$1 is irreducible and $E=E_1^k+E_k$2 for all $E=E_1^k+E_k$3.

Necessity is constructive: reducibility permits regrouping terms into an alternative representation, while any $E=E_1^k+E_k$4 allows either absorbing two copies into a doubled term elsewhere or inserting the new value $E=E_1^k+E_k$5.

Sufficiency requires substantially more work because no analogue of Banakiewicz's minimality characterization exists for semi-fast convergent sequences — the authors state this explicitly as an open problem. Instead they develop a geometric argument centered on the longest dominating gap $E=E_1^k+E_k$6. The key lemma shows that any monotone representation $E=E_1^k+E_k$7 of $E=E_1^k+E_k$8 must agree with $E=E_1^k+E_k$9 on the first $k$0 coordinates. The proof proceeds by backward induction along indices, ruling out candidate values of $k$1 through a careful analysis of "pseudointervals" — translates of the tail achievement set $k$2 — and their internal gaps. Two structural facts drive the contradictions: an observation that among gaps of order at most $k$3, only the last can have length shorter than $k$4 without its successor having length exactly $k$5; and the domination of $k$6, which forbids translated tails from containing gaps of that length. Both cases $k$7 and $k$8 terminate in violations of irreducibility. An induction over the infinite sequence of dominating gaps then forces global agreement of representations.

The contrast between the two proofs is methodologically notable: the central case reduces to a metric invariant, while the symmetric case requires direct combinatorial geometry precisely because the invariant-theoretic route is unavailable.

Cantorvals: partial results

For Cantorvals the picture remains fragmentary. Prior to this work, the Guthrie–Nymann Cantorval $k$9 was the only known uniquely achievable Cantorval. The authors prove a new sufficient condition:

If $I_k=E_1^k+[0,r_k]$0, $I_k=E_1^k+[0,r_k]$1, and $I_k=E_1^k+[0,r_k]$2 for all $I_k=E_1^k+[0,r_k]$3, then $I_k=E_1^k+[0,r_k]$4 is uniquely achievable.

The proof tracks principal dominating gaps across representations: each even-indexed term $I_k=E_1^k+[0,r_k]$5 must appear in any alternative representation with matching remainder, and a monotonicity contradiction pins down the interleave exactly. This immediately covers every two-block multigeometric Cantorval $I_k=E_1^k+[0,r_k]$6, recovering the Guthrie–Nymann case and adding new examples such as $I_k=E_1^k+[0,r_k]$7 and $I_k=E_1^k+[0,r_k]$8.

Two limitations are acknowledged candidly. First, the condition applies to sequences whose geometry mirrors the Guthrie–Nymann pattern; the authors show via their generator theorem for Cantorvals that, within that family, the Guthrie–Nymann Cantorval is the only uniquely achievable member — every other parameter choice forces a doubling somewhere in the sequence. Second, the theorem applies to two particular non-multigeometric sequences (a perturbed binary sequence and a sequence alternating near-halvings) whose achievement sets' topological types are unknown; unique achievability is established without knowing whether the sets are Cantorvals at all.

The paper closes with necessary obstructions (doublings, triple repetitions, triples of doubly repeated terms preclude uniqueness) and poses the natural converse question: whether these obstructions are the only ones for Cantorvals.

Limitations and open questions

Three gaps are explicit in the paper. There is no characterization of minimal centers of distances for semi-fast convergent sequences, which blocked extending the central-case method and motivates the authors' first posed problem. For Cantorvals, all known uniquely achievable examples arise from the alternating-gap condition, and whether others exist is open. Finally, the proposed full characterization of Cantorval uniqueness via absence of doublings and repetitions remains unverified in both directions beyond the special cases treated here.

Conclusion

The paper converts several sufficient conditions from the literature into exact characterizations: for central Cantor sets, unique achievability is equivalent to the absence of adjacent doublings, decided by the center-of-distances invariant; for symmetric Cantor sets, it is equivalent to irreducibility with multiplicities at most two, proved by geometric analysis of dominating gaps and pseudointervals. For Cantorvals, the alternating-gap criterion supplies new uniquely achievable examples but leaves the general classification open, with the obstruction-based converse as the natural next target.