Counting versions for restricted and somewhat-restricted 3-term arithmetic progressions

Investigate the counting versions of restricted and somewhat-restricted 3-term arithmetic progression problems in F_p^n by determining whether a subset A ⊆ F_p^n of constant density δ > 0 must contain a constant (2^{Ω(1)}) fraction of all somewhat-restricted 3-term arithmetic progressions and a constant (2^{Ω(1)}) fraction of all restricted 3-term arithmetic progressions.

Background

The paper considers restricted variants of 3-term arithmetic progressions in finite fields, including ‘restricted’ (a ∈ {0,1}^n \ {0}) and ‘somewhat-restricted’ (a ∈ {0,1,2}^n \ {0}) patterns. While density upper bounds have been obtained using inverse-correlation techniques (Theorems 5.2 and 5.3), the counting versions—establishing that dense sets contain a constant fraction of such progressions—remain unresolved.

These counting questions interface with product-space models of correlation and could require new analytic or combinatorial tools beyond current inverse theorems.