Let $v$ be a unit vector in $R^k$. Let $d \in \mathbb{Z}$. Consider the plane $$ P = \{x \in R^k : v \cdot x = d \}. $$ Let $\mu$ be the surface measure on $P$. In other words, $\mu$ is the $(k-1)$-dimensional Hausdorff measure on $R^k$, but restricted to $P$.
Note $P$ is $Z^k$-periodic. Consider $P$ as a subset of the torus $R^k / Z^k \simeq [0,1]^k$. I'm interested in the Fourier coefficients of the measure $\mu$: $$ \hat{\mu}(\gamma) = \int_{[0,1]^k} e^{-2 \pi i \gamma \cdot x} d\mu(x). $$ More specifically, I'm interested in the case where $\gamma$ is non-zero and orthogonal to $v$.
Consider the special case $v=(1,0,\ldots,0)$. Then the integral becomes $$ \hat{\mu}(\gamma) = \int_{[0,1]^{k-1}} e^{-2 \pi i (\gamma_2,\ldots,\gamma_k) \cdot (x_2,\ldots,x_k)} dx_2 \cdots dx_k $$ If at least one of $\gamma_2,\ldots,\gamma_k$ is non-zero, then the integral is zero. In other words, if $\gamma$ is non-zero and orthogonal to $v$, then $\hat{\mu}(\gamma) = 0$.
For arbitrary $v$, I suspect that the same result is true.
But how can I prove it?