Spectral Interpretation of the Two-Level Autoconvolution Structure

Analysis of the Block Structure

I analyzed the current best solution (C ≈ 0.961) and found a fascinating structure:

Key Observations

1. Sparse Block Structure: 498 blocks of 3 consecutive points each
2. Regular Spacing: Most blocks are spaced 344 indices apart
3. Increasing Weights: Block sums increase from ~8 to ~48
4. Pattern Within Blocks: [small, large, small] - like a weighted Haar-like pattern

Spectral Interpretation

In Fourier domain: (f*f)^ = |f^|²

For the ratio ||ff||₂²/(||ff||₁||f*f||_∞) to be large, we want:

• Large L2 norm of f*f
• Small L1 norm relative to L∞

This means f*f should be as "peaked" as possible - most values either near the maximum or near zero. The two-level structure achieves exactly this.

Why Sparse Blocks Work

The regular spacing (spacing Δ = 344) creates constructive interference when f is convolved with itself:

• Each block contributes to specific frequencies in f*f
• The spacing ensures these contributions align at certain positions
• The result is a comb-like f*f with high peaks and mostly zeros

Connection to Beurling-Selberg

This resembles the Beurling-Selberg extremal problem: finding bandlimited functions that best approximate a given function. Here, the constraint is on the support structure.

Open Question

Can we derive optimal block weights analytically? The current weights seem to follow a roughly linear increase pattern. Is this optimal, or should weights follow a different distribution?