On the spectra of planar self-affine measures with p digits

For a prime number $p\ge 3$, let $A\in M_2\left(Z\right)$ be an expanding matrix and let $B\subset Z^2$ be a p-element digit set satisfying that$Z\left({\hat{\delta }}_B\right)=\bigcup _{j=1}^{p-1}(\frac{j}{p}\left(\begin{array}{c}\omega \\ \rho \end{array}\right)+Z^2),$ where $\{\rho ,\omega \}\subset \{1,\cdots ,p-1\}$ and $\gcd (\rho ,\omega )=1$. Here $Z\left({\hat{\delta }}_B\right)$ denotes the zero set of the function ${\hat{\delta }}_B$. The associated self-affine measure ${\mu }_{A,B}$ is generated by the iterated function system (IFS):${\{f_b:f_b(x)=A^{-1}(x+b),x\in R^2\}}_{b\in B}.$ In this paper, some equivalent conditions for the self-affine measure ${\mu }_{A,B}$ to be spectral are obtained. This extends the result of An, He and Tao [2].