David Kumallagov

David Kumallagov

The Burrows--Wheeler transform is usually viewed as a clustering transform: it tends to group equal letters into long runs. We study the opposite extremal regime, where the BWT output is completely unclustered, that is, has as many equal-letter runs as positions. Known results imply, on the one hand, that the number of runs in the BWT of a Lyndon word can increase by at most a factor of two, and, on the other hand, that over every alphabet of size at least three completely unclustered BWT images exist in every length. This leads to the extremal problem lying between these two facts. For \(k\ge3\), let \(U_k(n)\) be the minimum cyclic run number of a primitive necklace of length \(n\) whose BWT has \(n\) runs. We prove the universal lower bound \(U_k(n)\ge\lceil n/2\rceil\), reduce the sharpness problem for one-cycle BWT images \(L\) to the Hamming identity \[ \cruns(\BWT^{-1}(L))=\dH(L,\sort(L)), \] and develop a natural multiset-of-necklaces relaxation with an explicit constant-cycle correction. We compute the small values, including the exceptional value \(U_k(6)=4\), prove a parity obstruction for the Parikh vectors of sharp examples, and determine the multiset relaxation exactly. Finally, for every prime \(p\equiv5\pmod8\) for which \(2\) is a primitive root modulo \(p\), we prove sharpness in the adjacent lengths \(p-1\) and \(p\). Under the corresponding Artin-type infinitude hypothesis, this gives infinitely many adjacent sharp pairs.

David Kumallagov, Daniil Sizikov, Anton Zarubin

We develop a Galois descent approach to finite-field Fourier spectra over an arbitrary finite base field. Let $\mathbb K=\mathbb F_q$ and $\mathbb L=\mathbb F_{q^m}$. If a Fourier transform is applied to a $\mathbb K$-valued vector, then its spectrum is not an arbitrary element of $\mathbb L^n$: it satisfies the Frobenius consistency relation \[ V_s^q=V_{qs \bmod n}. \] We prove a general Galois-descent theorem for Fourier transforms on finite abelian groups, characterize the one-dimensional spectra as products of subfields indexed by $q$-cyclotomic classes, and show that the orbit-seed representation is optimal in base-field coordinates. For arbitrary vectors in $\mathbb L^n$, we study a two-stage representation $g=f+h$, where $f$ is class-consistent and $h$ is a residual. The residual optimization separates over cyclotomic classes. We give exact support minimization, a symbol weight enumerator for the class-consistent code, recovery guarantees, global covering-radius formulas, random residual tail bounds, and entropy-type lower bounds. We also discuss implementation consequences for trace decompositions, normal bases, canonical subfield embeddings, and sparse-polynomial residual backends.