The cyclic interval lemma for every odd modulus

Abstract

Let ℤₘ=ℤ/mℤ and call W(a,L)={a,a+1,…,a+L-1}, 0 ≤ L ≤ m, an arc. An arc W beats a set x ⊆ ℤₘ when |W△ x|<min(|W|,m-|W|). Fokam Souop and Bitjoka, studying the least dimension kₘᵢₙ(Cₘ) of a binary Cayley graph into which the cycle Cₘ embeds isometrically, isolated the assertion that for odd m every x ∉ {emptyset,ℤₘ} is beaten by some arc. They proved it in one regime, verified it by computer for all odd m ≤ 17, and left it as a conjecture; it is the first of the three problems their paper leaves open. We prove it for every odd m. The argument instantiates the hypothesis "no arc beats x" at exactly two arc lengths, d and d+1 where m=2d+1, which squeezes every length-d window of x to the same cardinality, and a single double count then forces |x| ∈ {0,m}. We also settle the case the source does not discuss: for m=2h the sets that no arc beats are exactly the sets invariant under the antipodal rotation i ↦ i+h, and there are exactly 2ʰ of them, 2ʰ-2 of them proper and nonempty. So the parity hypothesis is not an artefact of the proof. An independent exhaustive search over all 2ᵐ-2 proper nonempty subsets of ℤₘ confirms the odd statement for every odd m ≤ 31 and the count 2ʰ-2 for every even m ≤ 24. Every result below is machine-checked in Lean 4.

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