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If $b_{ij}\in[0,2^t-1]\cap\mathbb Z$ holds at every $i\in\{1,\dots,k'\}$ and $j\in\{1,\dots,k\}$ where $1\leq k'\leq k$ holds then there are $x_1,\dots,x_k\in\mathbb Z:\sum_{j=1}^kx_jb_{1i}=0\wedge\dots\wedge\sum_{j=1}^kx_jb_{k'j}=0$ with $\sum_{j=1}^{k}|x_j|\neq0$. Further restrict $b_{ij}$ to set $T_t(\beta,q)$ of coprime integers in $[0,2^t-1]\cap\mathbb Z$ with

  1. $b_{\max}-b_{\min}<\beta2^t$ at fixed $\beta\in(0,1)$ and

  2. $\sum_{j=1}^kb_{ij}^2=q$ at every $i\in\{1,\dots,k'\}$.

$B$ is $k'\times k$ matrix of $b_{ij}$.

If $N(B)=\{|(x_1,\dots,x_k)|_\infty:\sum_{i=1}^kx_ib_{1i}=0\wedge\dots\wedge\sum_{i=1}^kx_ib_{k'i}=0\}$ then a good upper bound for $$\alpha(k,t,\beta)=\max_{b_{ij}\in T_t(\beta,q)}\min_{n\in N(B)}n$$ is given by the Bombieri-Vaaler lemma of $O((det(BB'))^{\frac1{2(k-k')}})$ where $B'$ is transpose of $B$.

  1. If $$N_q(B)=\{|(x_1,\dots,x_k)|_\infty:\sum_{i=1}^kx_ib_{1i}=0\wedge\dots\wedge\sum_{i=1}^kx_ib_{k'i}=0\wedge\sum_{i=1}^kx_i^2=q\}$$ where entries $b_{ij}$ of $B$ come from $T_t(\beta,q)$ and $N_q(B)\neq\emptyset$ holds then what is a good upper and lower bound for $q$?

  2. Is there a name for this $q$?

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    $\begingroup$ Probably you want $a_i\ne 0$, or at least not all of them 0. $\endgroup$ – Brendan McKay May 31 at 8:38

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