New optima for the deletion shadow

TL;DR

The paper addresses identifying minimal deletion shadows for large alphabet word families by proving that, for $s\le r$ and $k\le n$, the optimal discrete family is the set of words in $[s]^n$ in which the symbol $s$ occurs at most $k$ times. The authors lift the problem to a fractional setting with weights in $[0,1]$, define a consistent deletion shadow, and introduce $s$-compression and fractional Hamming balls $b^{(n,s)}_{k,\alpha}$ that are shown to minimize the shadow. The main fractional result states that among all $s$-compressed fractional $f$ with $|f|\le s^n$, the shadow is minimized by the corresponding fractional Hamming ball $h$, which implies the discrete extremal family via down-compression. The approach blends a fractional isoperimetric framework with a layered, case-by-case analysis (Cases 1–3) to establish optimality, and recovers the discrete minimal-shadow result for a dense range of sizes between the known $[s]^n$ and $(s+1)^n$ cases. These fractional techniques offer potential broader applications in isoperimetric-type problems on words and finite alphabets.

Abstract

For a family $\mathcal{F}$ of words of length $n$ drawn from an alphabet $A=[r]=\{1,\dots,r\}$, Danh and Daykin defined the deletion shadow $Δ\mathcal{F}$ as the family containing all words that can be made by deleting one letter of a word of $\mathcal{F}$. They asked, given the size of such a family, how small its deletion shadow can be, and answered this with a Kruskal-Katona type result when the alphabet has size $2$. However, Leck showed that no ordering can give such a result for larger alphabets. The minimal shadow has been known for families of size $s^n$, where the optimal family has form $[s]^n$. We give the minimal shadow for many intermediate sizes between these levels, showing that families of the form 'all words in $[s]^n$ in which the symbol $s$ appears at most $k$ times' are optimal. Our proof uses some fractional techniques that may be of independent interest.

Theorems & Definitions (8)

• Theorem 1.1: danh_ordering_1997
• Theorem 1.2
• Lemma 3.1
• proof : Proof of Lemma \ref{['lem:downcompression']}
• Theorem 4.1
• proof : Proof of Theorem \ref{['thm:discreteresult']} from Theorem \ref{['thm:fractionalresult']}
• proof : Proof of Theorem \ref{['thm:fractionalresult']}
• Conjecture 5.1