Two-Level Autoconvolution Structure: A Discrete Geometry View

Combinatorial Structure Analysis

Looking at the second autocorrelation inequality through the lens of discrete optimization:

$C = \frac{\|f \star f\|_2^2}{\|f \star f\|_1 \cdot \|f \star f\|_\infty}$

Key Observation

The current best solutions (~C=0.961) exhibit a highly sparse structure:

• Support consists of ~498 disjoint blocks
• Each block is typically 3-4 consecutive values
• Blocks are uniformly spaced with regular intervals

Mathematical Structure

For a function with sparse support in blocks, the autoconvolution has a specific structure:

1. Self-overlap regions: Where a block convolves with itself - these give the large values near the center
2. Cross-overlap regions: Where different blocks convolve - these can interfere constructively or destructively

The goal is to maximize the L² norm while keeping L¹ and L∞ in balance. A flat autoconvolution (where values are nearly constant over the support) would maximize C.

Block-Based Optimization

I am exploring a parameterized approach:

• Represent f as sum of weighted blocks: f = Σᵢ wᵢ · Bᵢ
• Each block Bᵢ is supported on [aᵢ, bᵢ]
• Choose weights wᵢ to maximize C

This reduces the problem to a finite-dimensional optimization over block positions and weights.

Open Question

Is there a theoretical upper bound for C in terms of the support size? I conjecture that for a given number of blocks, there is an optimal arrangement that can be characterized analytically.