Difference Sets of Signed Representatives

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$ with $n\geq 1$, and write $[n]=\{1,\ldots,n\}$. Suppose that $A\subseteq{\mathbb{Z}}$ satisfies \[ \left|A\cap\{k,-k\}\right|=1\qquad\text{for every }k\in[n]. \] Define \[ A-A=\{a-b:a,b\in A\},\qquad D_n(A)=(A-A)\cap[n]. \] For $B\subseteq[n]$, let $\mathbf 1_B(t)$ equal $1$ when $t\in B$ and $0$ otherwise, and define the ordered-pair count \[ R_A(B)=\sum_{(a,b)\in A\times A}\mathbf 1_B(a-b). \] The literal source condition permits elements of $A$ outside $\{-n,\ldots,-1,1,\ldots,n\}$, including infinitely many elements. For this literal reading, interpret the nonnegative sum as taking values in ${\mathbb{N}}\cup\{0,+\infty\}$. Alternatively, one may restrict attention to the finite set $A_0=A\cap\{-n,\ldots,-1,1,\ldots,n\}$ of signed representatives. Both proposed universal lower bounds are equivalent under these two readings, since $A_0\subseteq A$, and the finite signed-representative sets themselves satisfy the literal condition.

First question. Is it true, uniformly over all such $A$, that \[ |D_n(A)|=(1-o(1))n\qquad(n\to\infty)? \] Explicitly, does every $\eta\in(0,1)$ admit an integer $N_\eta\geq1$ such that, for every integer $n\geq N_\eta$ and every admissible $A$, \[ |D_n(A)|\geq(1-\eta)n? \]

Proposed stronger question. For every fixed $\varepsilon\in(0,1]$, do there exist $c_\varepsilon>0$ and an integer $N_\varepsilon\geq1$ such that, for every integer $n\geq N_\varepsilon$, every admissible $A$, and every $B\subseteq[n]$ with $|B|\geq\varepsilon n$, \[ R_A(B)\geq c_\varepsilon n^2? \] Here the constant and threshold may depend only on $\varepsilon$. This makes explicit the asymptotic reading of the supplied notation $\gg_\varepsilon n^2$.

The source remarks that replacing the alternatives $k,-k$ by $k,1-k$ permits a choice containing only even integers, so the analogous assertions fail. Its observation that the signed alternatives appear to have no congruence obstruction is a supplied heuristic qualification, not a proof. The original extract also contains an interleaved modular-forms passage referring to an unspecified equation (1) and reference [1]; its relationship to these questions is unresolved and it supplies no usable hypothesis for them.

Short English statement

Choose one sign for each integer from $1$ to $n$. Must the positive differences include all but $o(n)$ integers in $[n]$, uniformly over the choices? More strongly, must every subset of $[n]$ of fixed positive density receive at least a constant multiple of $n^2$ ordered difference representations? These are supplied open questions; this statement-only draft requires human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30006465$, title Difference Sets of Signed Representatives, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposer field is empty; the proposed year is $2025$.
• Supplied report citation: Analytic Number Theory (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/51, pp. 2701--2764. Supplied source URL and literature-source link, labelled "Oberwolfach report 51/2025": https://doi.org/10.4171/owr/2025/51. The apparent year discrepancy is retained as supplied. No specific problem page or individual proposer is supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels both assertions open and reports that the report notes no congruence obstruction but gives no proof. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that only formula delimiters, notation, grammar, and typography were repaired while mathematical content and scope were preserved. This supplied editorial label does not establish mathematical correctness or independent source verification.