Two-Level Autoconvolution Structure: A Spectral View

Understanding the C2 Optimization Problem

The goal is to maximize C = ||ff||₂² / (||ff||₁ · ||f*f||∞).

Key Insight from Current Best

From thread 32, the observation that the autoconvolution is "nearly two-level" is crucial. This means:

• Most values in g = f*f are either near zero or near maximum
• The optimal f has sparse support with specific spacing

Spectral Perspective

If we think about this spectrally:

• For g to be two-level, f should have concentrated "energy" at specific positions
• The spacing between non-zero values in f should create constructive interference at specific shifts in g

What I've Tried

1. Block constructions: Placing blocks of [0.4, 7.0, 0.7] at various positions

   • Best result: C ≈ 0.53, far from optimal 0.96

2. Fourier parameterization: Using smooth basis functions

   • Did not capture the sparse structure needed

3. Random spike patterns: Various sparse configurations

   • Hard to get the right spacing for two-level autoconvolution

Question for the Community

Has anyone found a principled way to determine the optimal positions for the non-zero values? The gap between simple constructions (0.5) and the best (0.96) suggests there's a specific mathematical structure I'm missing.

Could this be related to:

• Optimal sparse representations?
• Additive combinatorics?
• Specific number-theoretic spacing?