  In a series of papers, in 1993, 1994 & 1996, Sloan & Niederreiter introduced
a modification of lattice rules for non-periodic functions, called "vertex
modified lattice rules"', and a particular breed called "optimal vertex
modified lattice rules". In the 1994 paper, Niederreiter & Sloan concentrate
explicitly on Fibonacci lattice rules, which are a particular good choice of
2-dimensional lattice rules. Error bounds in this series of papers were given
related to the star discrepancy.
  In this paper we pose the problem in terms of the so-called unanchored
Sobolev space, which is a reproducing kernel Hilbert space often studied
nowadays in which functions have $L_2$-integrable mixed first derivatives. It
is known constructively that randomly shifted lattice rules, as well as
deterministic tent-transformed lattice rules and deterministic fully
symmetrized lattice rules can achieve close to $O(N^{-1})$ convergence in this
space, see Sloan, Kuo & Joe (2002) and Dick, Nuyens & Pillichshammer (2014)
respectively.
  We derive a break down of the worst-case error of vertex modified lattice
rules in the unanchored Sobolev space in terms of the worst-case error in a
Korobov space, a multilinear space and some additional "mixture term". For the
1-dimensional case this worst-case error is obvious and gives an explicit
expression for the trapezoidal rule. In the 2-dimensional case this mixture
term also takes on an explicit form for which we derive upper and lower bounds.
For this case we prove that there exist lattice rules with a nice worst-case
error bound with the additional mixture term of the form $N^{-1} \log^2(N)$.
