Given $a,m,n,t\in\Bbb Z$, with $n=m^t$ and $a$ arbitrary, and given $\mathbb{Z}$-linearly independent vectors $v_1,\dots,v_n\in\Bbb Z^n$, and an arbitrary vector $w\in\Bbb Z^n$, such that $$\langle v_i-v_j,w\rangle=0$$ for all $1\leq i<j\leq n$, are there technical terms for the set of vectors

$$\mathcal T_a=\{v\in\Bbb Q^n:\langle(\langle v_1,v\rangle,\langle v_2,v\rangle,\dots,\langle v_n,v\rangle),w\rangle=a\}?$$

$$\mathcal U_t=\{v\in\Bbb Q^n:\forall\ 1\leq i\leq t \exists u_i\in\Bbb Z^{m}\mbox{ with }(\langle v_1,v\rangle,\dots,\langle v_n,v\rangle)=u_1\otimes \dots\otimes u_t\}?$$

Moreover: is there a reasonable way to check if $\mathcal T_a\cap\mathcal U_t\neq\emptyset$?

puzzling: the condition ' $v=(\langle v_1,v\rangle,\langle v_2,v\rangle,\dots,\langle v_n,v\rangle)\in\Bbb Z^n$ ' is equivalent to $Mv=v$, where $M$ is the (as per your hypothesis:full-rank) matrix which has the $v_i$ for it rows; but row-rank equals column-rank, also for linear algebra over $\mathbb{Z}$, so this 'matrix equation' is non-contradictoryonlyif $M$ is theidentity matrix, i.e., $v_i=e_i$=the $i$-th standard basis vector, whereupon your last line becomes tautologous. Would you please clarify what you mean? $\endgroup$ – Peter Heinig Sep 30 '17 at 20:30formulationof the OP: it seems to me that the 'parameter' " $a$ " is totally redundant and (to me) confusing. The given data are $v_1,\dotsc, v_n,w\in\mathbb{Z}^n$, and $b\in\mathbb{Z}$, subject to the 'axiom' that the $v_i$ be linearly independent. The equation containing '$a$' is a tautology/does-not-impose-any-condition. It seems to me that the '$a$' can be eliminated from the context for good, right? $\endgroup$ – Peter Heinig Oct 1 '17 at 16:33equivalentto ' $\color{blue}{\forall i,j\in\{1,..,n\}\quad \langle v_i-v_j,w \rangle = 0}$ ' , the latter being the form I would strongly recommend to write your condition in, removing the '$a$' completely from the OP. $\endgroup$ – Peter Heinig Oct 1 '17 at 17:05