[Paper Review] Is the Multiset of $n$ Integers Uniquely Determined by the Multiset of Its $s$-sums?

Dmitri Fomin|arXiv (Cornell University)|Sep 18, 2017

Analytic Number Theory Research6 references3 citations

TL;DR

This paper investigates whether a multiset of $ n $ integers is uniquely recoverable from the multiset of its $ s $-sums—i.e., all possible sums of $ s $ distinct elements. Using symmetric polynomials, generating functions, and combinatorial constructions, the authors establish that recovery is impossible (i.e., $ s $-equivalent distinct multisets exist) if and only if $ n $ is a power of 2 when $ s = 2 $, and extend this to higher $ s $, revealing deep connections to algebraic number theory and combinatorics.

ABSTRACT

In 1957 Leo Moser published a problem in American Mathematical Monthly asking whether knowing the set of all pairwise sums of five numbers one could determine the original numbers. Problem was quickly generalized as "Is it always possible to restore a collection of $n$ numbers from the collection of its $s$-sums?"; it turned out to be a very interesting and nontrivial question in additive number theory and combinatorics. On its sixtieth anniversary we present here a survey of all the known research, results and techniques used in various attempts to solve this problem. Some new findings and open questions are presented as well.

Motivation & Objective

• To determine under what conditions a multiset of $ n $ integers is uniquely recoverable from the multiset of its $ s $-sums.
• To characterize all pairs $ (n, s) $ for which recovery fails—i.e., where distinct multisets produce identical $ s $-sum multisets.
• To unify and survey six decades of research on the multiset recovery problem, including foundational results and open questions.
• To present new findings on $ s $-equivalence and singular pairs, particularly for $ s > 2 $, using algebraic and combinatorial tools.
• To provide a comprehensive overview of the problem’s history, key theorems, and unresolved conjectures in number theory and combinatorics.

Proposed method

• Employing symmetric polynomials and generating functions to analyze the algebraic structure of $ s $-sum multisets.
• Using the theory of elementary symmetric polynomials to relate the multiset $ A $ to its $ s $-sum multiset $ A^{(s)} $, enabling reconstruction under certain conditions.
• Applying the concept of $ s $-equivalence: two multisets $ A $ and $ B $ are $ s $-equivalent if $ A^{(s)} = B^{(s)} $, and analyzing when such non-uniqueness occurs.
• Leveraging the Lambek-Moser theorem and complementary sequences to explore structural parallels in multiset recovery.
• Constructing explicit counterexamples for $ n = 2^k $, $ s = 2 $, showing that recovery fails precisely when $ n $ is a power of 2.
• Extending results beyond $ s = 2 $ using generating function identities and properties of roots of unity to analyze higher-order $ s $-sums.

Experimental results

Research questions

• RQ1 For which pairs $ (n, s) $ is a multiset of $ n $ integers uniquely determined by its $ s $-sum multiset?
• RQ2 What is the complete characterization of singular pairs $ (n, s) $, where $ n > s $ and $ A^{(s)} = B^{(s)} $ for distinct multisets $ A $ and $ B $?
• RQ3 Can the multiset recovery problem be solved for $ s > 2 $, and what algebraic or combinatorial invariants govern such recovery?
• RQ4 What is the role of symmetric polynomials and generating functions in determining the uniqueness of multiset reconstruction?
• RQ5 Are there structural or number-theoretic obstructions to recovery beyond the case $ s = 2 $, and how do they generalize?

Key findings

• For $ s = 2 $, the multiset $ A $ is uniquely recoverable from $ A^{(2)} $ if and only if $ n $ is not a power of 2, i.e., $ inom{n}{2} $ pairwise sums determine $ A $ uniquely unless $ n = 2^k $ for $ k > 1 $.
• The pair $ (n, 2) $ is singular if and only if $ n $ is a power of 2, with $ inname{M}_2 = \{4, 8, 16, 32, \dots\} $, confirming Selfridge and Straus’s 1958 result.
• For $ s > 2 $, the problem remains open in general, but the paper identifies structural conditions under which $ s $-equivalence can occur, particularly via roots of unity and symmetric function identities.
• The paper constructs explicit $ s $-equivalent pairs for $ s = 2 $, $ n = 4 $, showing that four numbers cannot be uniquely recovered from their six pairwise sums.
• The use of generating functions and elementary symmetric polynomials allows the derivation of necessary and sufficient conditions for multiset recovery in terms of polynomial roots.
• The paper highlights that the multiset recovery problem is deeply connected to algebraic number theory, particularly through the study of symmetric polynomials and their invariance under permutation of roots.