Proceedings of the National Academy of Sciences· 2026Q1
The largest support to escape chaos in random multiplicative functions
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- Q1SCImago
- 2026year
Short summary
A new result shows that for a random multiplicative function f(n), the convergence to a complex normal distribution requires the size of a subset A to be significantly smaller than N (specifically, |A| = o(N)). This condition is proven to be sharp, as a Gaussian limit holds for sets with density ρ where (1-ρ)^-1 = o((log log N)^1/2).
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Key points
- Convergence of |A|^-1/2 ∑_{n∈A} f(n) to a standard complex normal distribution forces |A| = o(N).
- The condition |A| = o(N) is sharp: a complex Gaussian limit holds for sets A of density ρ where (1-ρ)^-1 = o((log log N)^1/2).
- For positive density ρ, the correct normalization is (1-ρ)|A|.
- The factor (1-ρ) is nontrivial precisely when ρ does not tend to zero.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let f ( n ) be a Steinhaus random multiplicative function and let A ⊂ [ 1 , N ] be a finite set of integers. We show that convergence of | A | − 1 / 2 ∑ n ∈ A f ( n ) to a standard complex normal distribution forces | A | = o ( N ) . We prove that this zero-density condition is sharp: For most sets A of density ρ , and hence for some such sets, a complex Gaussian limit holds whenever ( 1 − ρ ) − 1 = o ( ( log log N ) 1 / 2 ) . However, for positive density, the correct normalization is ( 1 − ρ ) | A | rather than | A | . Thus, the additional factor 1 − ρ is nontrivial precisely when the density does not tend to zero.
The authors' abstract, as published at the source. Proceedings of the National Academy of Sciences, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics