A local mirrored-shadow law for sum representation functions

Definitions and mathematical statement

For finite $A,B\subseteq{\mathbb{Z}}$, define the representation count \[ \nu(n)=\#\{(a,b)\in A\times B:a+b=n\}=(\mathbf1_A*\mathbf1_B)(n). \] It has total mass $|A||B|$. Sort its positive values as $r_1\ge r_2\ge\cdots$, extending by zeros. These are the column heights of its Ferrers diagram. The corrected source condition is \[ r_k^2\le\sum_{j\ge k}r_j\qquad(k\ge1). \] V. Lev's question asks for other general constraints on integer-set representation functions, beyond that sorted-spectrum condition. For row lengths $N_t=\#\{n:\nu(n)\ge t\}$, the source also attributes to Pollard \[ \sum_{i=1}^tN_i\ge t(|A|+|B|-t),\qquad1\le t\le\min(|A|,|B|), \] and says it follows as well from the column condition. Missing cardinality signs and the split square in the extraction are corrected explicitly.

The attached aid gives a location-sensitive condition. For each $n$ let $R_n=A\cap(n-B)$ and, for $d\ge1$, let \[ q_n(d)=\#\{a\in R_n:a+d\in R_n\}. \] Its stronger fiberwise bounds and their sum are \[ \nu(n-d)\ge q_n(d),\qquad\nu(n+d)\ge q_n(d), \] \[ \sum_{d\ge1}\min\{\nu(n-d),\nu(n+d)\}\ge\binom{\nu(n)}2. \] All sums are finite. Each positive-difference fiber pair produces a distinct crossed representation at both reflected locations.

For every finite $R\subset{\mathbb{Z}}$, the choice $A=R$, $B=n-R$ realizes equality with the full difference profile: $\nu(n)=|R|$ and $\nu(n-d)=\nu(n+d)=\#\{a\in R:a+d\in R\}$. Thus the condition is sharp for all fiber sizes and realized positive-difference profiles. It contains information lost by sorting. For example the finitely supported proposed function with values $\nu(0)=2$, $\nu(1)=\nu(2)=1$ has sorted profile $(2,1,1)$ satisfying every column inequality, but violates the mirrored inequality at $0$. The aid provides a new necessary condition, not a complete characterization of all realizable representation functions.

Short English statement

Find constraints on additive representation functions beyond their sorted Ferrers diagrams. The supplied aid gives sharp reflected-location inequalities from each central representation fiber, retaining positional information that sorting discards.

Sources

• Local UnsolvedMath registry, permanent ID $20001102$, source code catalogue entry; canonical aim-combinatorics-notes.json, notes[226].
• American Institute of Mathematics, Recent trends in additive combinatorics, collected by Ernie Croot and Vsevolod F. Lev, problem 2.2 attributed to V. Lev: https://aimath.org/WWN/additivecomb/additivecomb.pdf. The source names Pollard without a complete citation. The page header and corrupted display are extraction artifacts; no PDF inspection is claimed here.
• Attached research attempt 1: partial result, recorded run status open, rigor proved, novelty confidence low; registry status partially solved, classification PARTIAL-PROGRESS. Its mirrored-shadow law is machine generated and not peer reviewed; independently verify before citation. No authoritative bibliography for the candidate inequality is supplied.