The difference between consecutive members of a difference set (𝑚𝑜𝑑 𝑣)

Abstract

Let D = { d 1 , d 2 , ⋯ , d k } \mathcal {D} = \{ {d_1},{d_2}, \cdots ,{d_k}\} be a difference set ( mod v ) \pmod v so that for any d ≢ 0 ( mod v ) d\not \equiv 0\pmod v there are exactly λ \lambda pairs ( d i , d j ) , d i , d j ∈ D ({d_i},{d_j}),{d_i},{d_j}\in \mathcal {D} such that d i − d j ≡ d ( mod v ) {d_i} - {d_j} \equiv d\pmod v . Suppose further that 0 ≤ d 1 > d 2 > ⋯ > d k > v 0 \leq {d_1} > {d_2} > \cdots > {d_k} > v and write d k + 1 = v + d 1 {d_{k + 1}} = v+ {d_1} . The following two results are proved: (i) Σ i = 1 k ( d i + 1 − d i ) 2 = O ( ( v 2 / k ) log ⁡ k ) , \Sigma _{i = 1}^k{({d_{i + 1}} - {d_i})^2} = O(({v^2}/k)\log k), (ii) max 1 ⩽ i ⩽ k ( d i + 1 − d i ) = O ( v / √ k ) . {\max _{1 \leqslant i \leqslant k}}({d_{i + 1}} - {d_i}) = O(v/\surd k).