The paper deals with the oscillation properties of the linear delay difference equation \[ \Delta x_{n}+p_{n}x_{n-k}=0, \tag{1} \] where \(\Delta\) is the forward difference operator, i.e., \(\Delta x_{n}=x_{n+1}-x_{n}\), and \(\{p_{n}\}\) is a sequence of nonnegative numbers. Under the critical condition \[ \liminf_{n\to\infty}p_{n}=\frac{k^{k}}{(k+1)^{k+1}} \] the authors establish that if \(\displaystyle a_{n}=p_{n}-\frac{k^{k}}{(k+1)^{k+1}}\geq 0\) for sufficiently large \(n\), then equation (1) is oscillatory if and only if the difference equation \[ \Delta^{2}y_{n-1}+2\frac{(k+1)^{k}}{k^{k+1}}a_{n}y_{n}=0 \] is oscillatory.
As a corollary some oscillation criteria for equation (1) are derived.