A weirdly constructed set of mathematical functions has disproven two of Paul Erdős's most famous predictions about how numbers behave.
April 23, 2026
Original Paper
Counterexamples for lacunary dilates via dyadic spike blocks
arXiv · 2604.18535
The Takeaway
Paul Erdős believed that when you take sparse samples of a mathematical function and average them out, the result should eventually behave in a predictable and bounded way. These specific lacunary averages were thought to represent a fundamental bridge between randomness and structure in harmonic analysis. New mathematical counterexamples using dyadic spike blocks demonstrate that these averages can actually explode and fail to converge where they were expected to stay stable. The construction reveals that the geometric gaps between these samples are not enough to guarantee the kind of law of large numbers behavior that the field relied on for decades. This discovery wipes out two major conjectures that guided the mathematical understanding of how signals and waves stabilize over time. Future signal processing algorithms will need to account for these instabilities to ensure that data sampled at geometric intervals does not produce false results.
From the abstract
We construct lacunary counterexamples for two problems of Erdős on the pointwise behavior of dilates $f(n_jx)$ on the circle. The method is a stagewise dyadic spike-block construction: each stage adds a small $L^2$ block and inserts many independent lacunary trials arranged so that a hit on one central spike creates a large partial average, while separation of binary scales keeps the old, active, and future blocks under control.First, we show that very weak Fourier-tail assumptions do not ensure