Exact quadratic-residue half-set counts and a quantitative Varnavides transfer

Definitions and mathematical statement

Let $p$ be an odd prime and $A\subseteq\mathbb F_p$. Count nonconstant three-term progressions by \[ T_p(A)=\left|\{(a,d)\in\mathbb F_p\times\mathbb F_p^*: a,a+d,a+2d\in A\}\right|. \] All arithmetic takes place modulo $p$. The common difference $d$ is nonzero, so all three terms are distinct. This is a count of ordered progressions: both directions are counted, and in the small field $\mathbb F_3$ a three-element set has more than one choice of middle term. Defining the count by $(a,d)$ removes such multiplicity ambiguities. Constant progressions are excluded. For an integer $0\le m\le p$, put \[ M(p,m)=\min_{|A|=m}T_p(A). \] The first question is to determine $M(p,(p-1)/2)$, or obtain its sharp asymptotic behaviour as $p\to\infty$ through odd primes. The second asks the corresponding question for density below one half: for fixed $0<\delta<1/2$, determine $M(p,\lfloor\delta p\rfloor)$, including the dependence on $\delta$. The floor makes the requested cardinality integral without changing its limiting density. A useful normalized target is $M(p,\lfloor\delta p\rfloor)/p^2$; the workshop remarks specifically motivate understanding its small $\delta$ behaviour. All subsets of the prescribed size are eligible, without algebraic or symmetry restrictions, so calculations for one family alone do not identify the minimum.

Short English statement

Among subsets of a prime field of a prescribed density at or below one half, how few nonconstant three-term progressions can occur?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001085 (AIM-COMBINATORICS-0210), 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 1.4. Original workshop question underlying AIM-COMBINATORICS-0210. https://aimath.org/WWN/additivecomb/additivecomb.pdf.