# Geometric-pattern avoidance research construction

Family 084: The geometric case of the Erdős similarity conjecture. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Combinatorial-analysis researchers can study positive-measure sets excluding every affine copy of a geometric progression.

## Applicability and commercial boundary

Near-full measure does not provide an efficient digital mask or a general privacy/anomaly algorithm; infinite affine-copy avoidance differs from finite patterns.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Inspect explicit construction and finite approximation costs before a visualization.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and selected formal statement have not yet been compared in depth for this family. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[The geometric case of the Erdős similarity conjecture](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/The-geometric-case-of-the-Erdos-similarity-conjecture-October-5-2026/geometric-erdos-similarity.pdf).