[Paper Review] On the determination of sets by their triple correlation in finite cyclic groups

Tamás Keleti, Mihail N. Kolountzakis|ArXiv.org|Mar 16, 2006

Limits and Structures in Graph Theory12 references3 citations

TL;DR

This paper investigates the extent to which a subset of a finite cyclic group ℤₙ is uniquely determined by its triple correlation (3-deck), showing that for odd n, a random subset is almost surely uniquely determined up to translation by its 3-deck. Using Fourier analysis and probabilistic bounds on the Fourier coefficients of random sets, the authors establish an exponentially small upper bound on the probability of non-uniqueness, significantly improving upon the previous O(1/√n) bound.

ABSTRACT

Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we know for all subsets $T$ of $G$ of size up to $k$ for how many $x \in G$ the translate $x+T$ is contained in $E$. This information is collectively called the $k$-deck of $E$. One can naturally extend the domain of definition of the $k$-deck to include functions on $G$. Given the group $G$ when is the $k$-deck of a set in $G$ sufficient to determine the set up to translation? The 2-deck is not sufficient (even when we allow for reflection of the set, which does not change the 2-deck) and the first interesting case is $k=3$. We further restrict $G$ to be cyclic and determine the values of $n$ for which the 3-deck of a subset of $\ZZ_n$ is sufficient to determine the set up to translation. This completes the work begun by Grünbaum and Moore as far as the 3-deck is concerned. We additionally estimate from above the probability that for a random subset of $\ZZ_n$ there exists another subset, not a translate of the first, with the same 3-deck. We give an exponentially small upper bound when the previously known one was $O(1\bigl / \sqrt{n})$.

Motivation & Objective

• To determine for which finite cyclic groups ℤₙ the 3-deck uniquely determines a subset up to translation.
• To improve upon the known O(1/√n) upper bound for the probability that a random subset is not uniquely determined by its 3-deck.
• To extend the work of Grünbaum and Moore on the 3-deck problem in finite abelian groups, particularly for cyclic groups.
• To use Fourier analytic techniques and probabilistic methods to estimate the likelihood of non-uniqueness in 3-deck reconstruction.

Proposed method

• The 3-deck is defined as the triple correlation function N_E,3(x,y) = ∑_z χ_E(z)χ_E(z+x)χ_E(z+y), counting translates of a 3-point pattern in E.
• Fourier transform techniques are used to relate the 3-deck to the magnitude of the Fourier transform of the indicator function χ_E.
• The key identity N_f,3 ≡ N_g,3 ⟺ |f̂(ξ)| |f̂(η)| = |ĝ(ξ)| |ĝ(η)| for all ξ, η with ξ+η=0 is derived.
• Probabilistic bounds on the Fourier coefficients of random subsets are established using linear independence over ℚ of roots of unity and a lemma on random sums.
• The concept of an 'extendable domain' in the Fourier support is used to link the structure of the Fourier transform to uniqueness of the set.
• The proof combines bounds on the number of small divisors of n and the growth of the divisor function d(n) to refine the failure probability to 2^{-C_ε n^{1-ε}}.

Experimental results

Research questions

• RQ1 For which finite cyclic groups ℤₙ is the 3-deck sufficient to determine a subset up to translation?
• RQ2 What is the asymptotic probability that a random subset of ℤₙ is not uniquely determined by its 3-deck?
• RQ3 Can the previously known O(1/√n) upper bound on the non-uniqueness probability be improved?
• RQ4 How does the structure of the Fourier transform support of a set relate to its 3-deck uniqueness?
• RQ5 What role does the number of divisors d(n) play in bounding the failure probability for 3-deck reconstruction?

Key findings

• For odd n, the probability that a random subset of ℤₙ is not uniquely determined by its 3-deck is at most 2^{-Cn / log log n}.
• For any ε > 0, the probability that a random subset of ℤₙ is not uniquely determined by its 3-deck is at most 2^{-C_ε n^{1-ε}}.
• The bound 2^{-C_ε n^{1-ε}} improves significantly upon the prior O(1/√n) upper bound.
• If the Fourier support of χ_E contains {1, 2, ..., d(n)}, then E is uniquely determined by its 3-deck.
• The proof relies on showing that the Fourier support being large enough (containing the first d(n) integers) implies the support is an extendable domain, which ensures 3-deck uniqueness.
• The result is established via probabilistic bounds on the vanishing of Fourier coefficients of random sets, using linear independence of roots of unity over ℚ.