Transversal Difference Numbers in Finite Abelian Quotients

Mathematics > Number Theory [Submitted on 26 Jun 2026] Transversal Difference Numbers in Finite Abelian Quotients Mugurel Barcau, Vicenţiu Paşol, George C. Ţurcaş Given \(H\leq G\) finite abelian groups, a transversal \(T\subseteq G\) for \(G/H\) has fixed size \(|G/H|\), but its ambient difference support \(D(T)=T-T\) can vary with the embedding of \(H\) in \(G\). We call \delta(G,H)=\min_T |D(T)| the transversal difference number of the pair \((G,H)\). This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption. We prove various results regarding this invariant, including a general lower bound \delta(G,H)\geq 2|G/H|-m(G,H), where \(m(G,H)\) is the largest order of a subgroup of \(G\) disjoint from \(H\). The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors. These results isolate the first genuinely new residual obstruction, namely the same-prime square plane G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG. For odd \(p\), this case is the technical core of the paper. Here transversals are graphs of functions \(\mathbb F_p^2\to \mathbb F_p^2\), and \(D(T)\) decomposes into carry-corrected finite-field derivative images. We conjecture that \delta(G,H)=(2p-1)^2 for all odd primes \(p\), prove the unconditional lower bound \(3p^2-p-1\), and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture. Comments: 27 pages, comments welcome Subjects: Number Theory (math.NT); Cryptography and Security (cs.CR); Discrete Mathematics (cs.DM); Combinatorics (math.CO) MSC classes: 05B45, 05B10, 11T06 Cite as: arXiv:2606.27961 [math.NT]   (or arXiv:2606.27961v1 [math.NT] for this version)   https://doi.org/10.48550/arXiv.2606.27961 Focus to learn more Submission history From: George Cătălin Ţurcaş [view email] [v1] Fri, 26 Jun 2026 11:03:47 UTC (30 KB) Access Paper: HTML (experimental) view license Current browse context: math.NT < prev   |   next > new | recent | 2026-06 Change to browse by: cs cs.CR cs.DM math math.CO References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)