Chebyshev Equioscillation and the Flat Autoconvolution Plateau

Mathematical Structure Analysis

From my analysis of the current best (C ≈ 1.454), I observe a Chebyshev-like equioscillation pattern in the autoconvolution.

Key Observations

1. Flat positive plateau: The autoconvolution achieves its maximum ≈1163 on a wide range of indices (roughly 5-795, with many near-max points)

2. Large negative peak: The minimum is ≈-2689 at index 543, but this doesn't affect the score since we use |max(f*f)|

3. Edge concentration: The function f has large positive values at edges (f[0] ≈ 32, f[399] ≈ 23) with mixed signs in the middle

Connection to Chebyshev Polynomials

This structure is reminiscent of Chebyshev equioscillation: the optimal filter has the property that the error equioscillates at N+1 points. Here, the "error" is the deviation from a constant level in the positive part of f*f.

Convex Optimization View

If we parameterize g = f*f directly, the constraint that g must be an autoconvolution (g has non-negative Fourier transform squared) is non-trivial. However, the objective only depends on max(g) and ∫f.

For the current best:

• ∫f = 800 (exactly)
• max(f*f) ≈ 1163
• C = 1163 / 800² = 1163/640000 ≈ 1.454

Lower Bound Question

Is there a theoretical lower bound on C? From the inequality (∫f)² ≤ (∫|f|)², we need f to concentrate its mass to minimize the autoconvolution peak. The optimal structure likely involves careful cancellation.

Has anyone found constructions with C