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The uncertainty principle and a generalization of a theorem of Tao. (English) Zbl 1242.43008

T. Tao’s uncertainty principle [Math. Res. Lett. 12, No. 1, 121–127 (2005; Zbl 1080.42002)] states that if \(p\) is prime then, for the cyclic group \(\mathbb{Z}/p\) of integers modulo \(p\), the support of any function \(f\) on \(\mathbb{Z}/p\) and its group Fourier transform \(\widehat{f}\) must satisfy \[ |\text{supp}\, f|+|\text{supp}\, \widehat{f}|\geq p+1 \] where \(|\cdot|\) denotes the number of elements of a subset. The result boils down to nonvanishing of a certain Vandermonde determinant. In the present work this result is extended to the cyclic group \(\mathbb{Z}/m\) in which \(m\) is composite and the support of \(f\) has a certain structure property related to vanishing exponential sums.
To describe this property, recall that a dominant weight is a multi-index \(\kappa=(\kappa_1,\dots,\kappa_n)\) such that \(\kappa_i\geq \kappa_{i+1}\). By the Weyl character formula, each dominant weight \(\lambda\) corresponds to a unique irreducible character \(\chi_\lambda\) of the group \(U(n)\) of unitary \(n\times n\) matrices and \(\chi_\lambda\) is a function on the \(n\)-torus such that \(\chi_\lambda(1)={\prod_{1\leq i<j\leq n} (\kappa_i-\kappa_j)}/{\prod_{1\leq i<j\leq n} (j-i)}\) where \(\kappa=\lambda+\rho\), \(\rho=(n-1,n-2,\dots, 1,0)\).
It was proved by T. Y. Lam and K. H. Leung [J. Algebra 224, No. 1, 91–109 (2000; Zbl 1099.11510)] that if \(p_1,\dots, p_r\) are the prime factors of \(m\) then the set \(W(m)\) of integers \(n\geq 0\) such that there exists a vanishing sum \(\omega_1+\dots +\omega_n=0\), where each \(\omega_i\) is an \(m\)th root of unity, is precisely \(W(m)=\mathbb{N} p_1+\dots + \mathbb{N} p_r\).
The main result states that if the support of \(f\) is represented by integers \(\kappa_1,\dots, \kappa_n\) such that \(\chi_\lambda(1)\notin W(m)\) then \[ |\text{supp}\, f|+|\text{supp}\, \widehat{f}|\geq m+1\, . \] Conversely, if \(A\) and \(B\) are subsets of \(\mathbb{Z}/m\) such that \(A\) is represented as above and if \(B\) is such that \(|A|+|B|\geq m+1\) then there is a function \(f\) such that \(f\) is supported inside \(A\) and \(\widehat{f}\) is supported on \(B\).

MSC:

43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
42A99 Harmonic analysis in one variable
11B75 Other combinatorial number theory
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References:

[1] Fulton, W.; Harris, J., Representation Theory. A First Course, Graduate Texts in Mathematics, vol. 129 (2001), Springer-Verlag
[2] Lam, T. Y.; Leung, K. H., On vanishing sums of roots of unity, J. Algebra, 109, 91-109 (1995) · Zbl 1099.11510
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