An incomplete sums-differences prompt and an exact slope-height obstruction

Definitions and mathematical statement

The exact source, attributed to N. Katz, is only \[ \operatorname{SD}(r_1,\ldots,r_n;\alpha),\qquad \text{for some }r_1,\ldots,r_n\in{\mathbb{R}}. \] The supplied verification says the official PDF, page 12, and TeX contain the same fragment: there is no verb, no quantifier or target for $\alpha$, and no definition of $\operatorname{SD}$ in that source. Its status is unrecoverable; no unique well-formed original problem can be inferred.

The aid separately studies a historical sums--differences interpretation. The following explicit projection model describes its auxiliary certificates, not an authenticated expansion of the fragment. Let $\pi_r(x,y)=x+ry$ for real $r$, $\pi_\infty(x,y)=y$, and take $\pi_{-1}$ as the injective target projection. For coprime nonzero integers $a,b$, with $a\ne b$ and $q=\max(|a|,|b|)\ge2$, consider the scaled grid \[ \mathcal G=\{(bx,-ay):x,y\in\{0,\ldots,q-1\}\}. \] Its target projection is $bx+ay$, injective on this grid, while the input slopes $0,-b/a,\infty$ have projection cardinalities $q,2q-1,q$. The aid reports that side length $q$ is maximal among consecutive square grids for that target injectivity. Thus its set-exponent certificate has \[ \alpha_0=\frac{2\log q}{\log(2q-1)},\qquad |\mathcal G|=q^2=(2q-1)^{\alpha_0}. \] For a literal unit-constant inequality $|\mathcal G|\le(\max_r|\pi_r\mathcal G|)^\alpha$, this obstructs $\alpha<\alpha_0$, whereas equality at $\alpha=\alpha_0$ is not a violation. The aid states failure at all $\alpha\le\alpha_0$; without its exact historical $SD$ definition, strictness and constants, that inclusive endpoint cannot be verified or promoted to a precise theorem here. In particular, allowing an unspecified multiplicative constant changes what a single finite grid can refute.

For the uniform distribution on the grid, the reported entropy ratio is \[ \frac{H(\mathcal G)}{\max_{r\in\{0,-b/a,\infty\}}H(\pi_r\mathcal G)} =\frac{2\log q}{h_q}, \] where $h_q$ is the entropy of the sum of two independent uniform variables on $\{0,\ldots,q-1\}$. Explicitly, \[ h_q=-\sum_{t=0}^{2q-2}\frac{c_t}{q^2}\log\frac{c_t}{q^2},\qquad c_t=\begin{cases}t+1,&0\le t<q,\\2q-1-t,&q\le t\le2q-2.\end{cases} \] All logarithms use a common base. The projective slope $\infty$ used here is not among the literal real slopes in the original fragment; this is another reason to keep the auxiliary model distinct. These supplied certificates neither identify the original requested $\alpha$ nor solve an authenticated source conjecture.

Short English statement

The original entry is an unrecoverable fragment with no definition of $SD$ or requested exponent. The supplied aid adds a particular sums--differences projection model and grid/entropy certificates; the draft keeps that interpretation separate and flags its unresolved endpoint claim.

Sources

• Local UnsolvedMath registry, permanent ID $20001140$, source code catalogue entry; aim-combinatorics-notes.json, notes[264], tag problem.
• AIM workshop "Recent trends in additive combinatorics," Problem 4.9, N. Katz, page 12: https://aimath.org/WWN/additivecomb/additivecomb.pdf. Statement status unrecoverable. The supplied verification reports the same fragment in official PDF and TeX, without any local definition of $SD$. No PDF or TeX source was accessed for this draft.
• No original remarks, literature field, full historical sums--differences citation, or authoritative external reference for the auxiliary certificate is supplied.
• Research attempt 1: invalid_statement, problem status at run unknown, rigor label proved, novelty confidence low; registry status open, classification OPEN-TRIAGE, suggested difficulty L2. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.