The higher-dimensional generalization of the weighted \( q \)-adic
sum-of-digits function
\( s_{q,\gamma}(n) \), \( n=0,1,2,\ldots \),
covers several important cases of sequences investigated in the theory of
uniformly distributed sequences, e.g., \( d \)-dimensional van der Corput–Halton
or \( d \)-dimensional Kronecker sequences. We prove a necessary and sufficient
condition for the higher-dimensional weighted \( q \)-adic sum-of-digits
functions to be uniformly distributed modulo one in terms of a trigonometric
product.

As applications of our condition we prove some upper estimates of the extreme
discrepancies of such sequences, and that the existence of distribution
function

\[
g(x)=x
\]

implies the uniform distribution modulo one of the weighted
\( q \)-adic sum-of-digits function
\( s_{q,\gamma}(n) \),
\( n=0,1,2,\ldots \).
We also prove the uniform distribution modulo one of related sequences

\[
h_{1q,\gamma}(n)+h_{2sq,\gamma}(n+1),
\]

where \( h_1 \) and \( h_2 \) are integers such that
\( h_1+h_2 \neq 0 \)
and that the akin two-dimensional sequence

\[
\bigl(s_{q,\gamma}(n),\,s_{q,\gamma}(n+1)\bigr)
\]

cannot be uniformly distributed modulo one if
\( q\ge 3 \).
The properties of the two-dimensional sequence

\[
\bigl(s_{q,\gamma}(n),\,s_{q,\gamma}(n+1)\bigr),
\qquad n=0,1,2,\ldots,
\]

will be instrumental in the proofs of the main results.