Sliding progression-free windows via gap words

Definitions and mathematical statement

Fix an integer $s\ge3$. Let $A=\{a_1<a_2<\cdots\}$ be an infinite increasing sequence of nonnegative integers. Require that every block of $s$ consecutive terms, $\{a_{i+1},\ldots,a_{i+s}\}$ for $i\ge0$, contain no nontrivial three-term arithmetic progression. A nontrivial progression consists of distinct elements $x<y<z$ satisfying $x+z=2y$. The block is consecutive in the ordered list of members of $A$, rather than a block of $s$ consecutive ambient integers. Equivalently, a progression $a_i,a_j,a_k$ with $i<j<k$ is allowed only if $k-i\ge s$.

To specify density for sequences without a natural density, set \[ \overline d(A)=\limsup_{N\to\infty} \frac{|A\cap\{0,1,\ldots,N-1\}|}{N},\qquad \Delta_s=\sup_A\overline d(A), \] where the supremum runs over sequences satisfying the block condition. The problem is to determine $\Delta_s$ and describe the extremal sets attaining it. If density is required to exist as a limit, the parallel question uses that natural density instead; this distinction is stated explicitly because the source does not specify a convention. The given examples are the residue classes $0,1,3,4$ modulo $6$ for $s=4$, of density $2/3$, and $0,1,3,4,9,10,12,13$ modulo $18$ for $s=8$, of density $4/9$. These provide benchmarks, not an assertion that all extremizers are periodic.

Short English statement

How dense can an integer sequence be if each block of s consecutive sequence elements avoids three-term progressions, and what do extremizers look like?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001084 (AIM-COMBINATORICS-0209), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Recent trends in additive combinatorics, item 1.3. Original workshop question underlying AIM-COMBINATORICS-0209. https://aimath.org/WWN/additivecomb/additivecomb.pdf.