Use ternary digits to build a large set modulo 3ᵏ

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The eight numbers with ternary digits 0 or 1 are progression-free modulo 27 when terms must be distinct.

The ternary digit example modulo 27

The eight residues with ternary digits 0 or 1 form a set with no three distinct terms in arithmetic progression modulo 27.

What was checked

The set is {0, 1, 3, 4, 9, 10, 12, 13}. Every ordered pair of distinct elements was checked against the third term of a progression modulo 27.

What this establishes

This confirms one finite example. It does not prove an asymptotic density bound or a statement for every modulus.

A nearby failure

The same digit rule does not transfer directly to modulus 4. The residues (3, 0, 1) form a nontrivial progression.