Restricted Roth progressions and their local obstructions

Definitions and mathematical statement

Write $[N]=\{1,\ldots,N\}$, and let $\log$ denote the natural logarithm. A nonconstant three-term arithmetic progression in $[N]$ is a triple $x,x+d,x+2d$ with integers $x,d$, $d\geq1$, and $1\leq x<x+2d\leq N$. Here the permitted differences are the shifted primes $\mathcal D=\{p-1:p\text{ is prime}\}$; in particular, the prime $2$ gives the permitted difference $1$. All three terms must belong to the same set, and they are distinct.

The conjecture asks whether, for every real $C>0$, there exists an integer $N_0(C)$ such that for every $N\geq N_0(C)$ and every $A\subseteq[N]$, \[ |A|\geq\frac{N}{(\log N)^C} \quad\Longrightarrow\quad \exists\,p\text{ prime},\ x\in\mathbb Z: \{x,x+p-1,x+2(p-1)\}\subseteq A. \] Equivalently, if $r_{p-1}(N)$ denotes the largest cardinality of a subset of $[N]$ avoiding all these triples, prove $r_{p-1}(N)<N/(\log N)^C$ for every fixed $C>0$ once $N$ is sufficiently large. The threshold may depend on $C$, but the conclusion must hold uniformly over the sets $A$. The restriction is on the common difference, not on the primality of the three terms themselves. This is the prime-minus-one question from the workshop; variations involving sums of two squares are separate questions and do not alter the specified difference set.

Short English statement

Does every set denser than any fixed inverse power of the logarithm contain a three-term progression whose difference is one less than a prime?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000880 (AIM-COMBINATORICS-0005), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, High-dimensional phenomena in discrete analysis, item 2.2. Original workshop question underlying AIM-COMBINATORICS-0005. http://aimpl.org/highdimdiscrete/2/.
• S3 Ben Green, Open problems, problem 11(ii). Quantitative shifted-prime progression context. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.