# Typical size of $\infty$ norm of integer points in subspaces associated to a structured linear diophantine equation

Take natural numbers $$A_1,B_1,A_2,B_2$$ random pairwise coprime in $$[n,2n]$$ for $$n$$ large enough and consider the space of solutions to $$A_1a+B_1b=0$$ and $$A_1A_2a+A_1B_2b+B_1A_2c+B_1B_2d=0$$ spanned by $$1\times 2$$ and $$3\times 4$$ matrices $$N_1=\begin{bmatrix} -B_1&A_1 \end{bmatrix}$$ and $$N_2=\begin{bmatrix} -B_2&A_2&0&0\\ -B_1&0&A_1&0\\ 0&0&-B_2&A_2 \end{bmatrix}$$ respectively.

Denote the $$\Bbb Q$$-linear space spanned by rows of $$N_1$$ by $$T_{A_1,B_1}\subseteq\Bbb Q^2$$ and $$N_2$$ by $$T_{A_1,B_1,A_2,B_2}\subseteq\Bbb Q^4$$ and denote the set of non-zero $$\Bbb Z$$ vectors in $$T_{A_1,B_1}$$ by $$T_{A_1,B_1}[\Bbb Z]^\star$$ and in $$T_{A_1,B_1,A_2,B_2}$$ by $$T_{A_1,B_1,A_2,B_2}[\Bbb Z]^\star$$.

Consider the quantities $$\mu_1((A_1,B_1))=\min_{v\in T_{A_1,B_1}[\Bbb Z]^\star}\|v\|_\infty$$ $$\mu_2((A_1,B_1)\otimes(A_2,B_2))=\min_{v\in T_{A_1,B_1,A_2,B_2}[\Bbb Z]^\star}\|v\|_\infty$$ where $$\|v\|_\infty$$ is largest coordinate by magnitude of vector $$v$$. One can show with probability $$1-o(1)$$ value of $$\mu((A_1,B_1))$$ is at least $$c_1\cdot n$$ for a constant $$c_1>0$$ and of $$\mu_2((A_1,B_1)\otimes(A_2,B_2))$$ is at least $$c_2\cdot n^{2/3}$$ for a constant $$c_2>0$$.

Similarly define $$\mu_k((A_1,B_1)\otimes(A_2,B_2)\otimes\dots\otimes(A_k,B_k))$$ where $$A_1,B_1,A_2,B_2,\dots,A_k,B_k$$ are random pairwise coprime in $$[n,2n]$$ for $$n$$ large enough.

It is possible to show that the typical value of $$\mu_k((A_1,B_1)\otimes(A_2,B_2)\otimes\dots\otimes(A_k,B_k))$$ is at least $$c_k\cdot n^{k/(2^k-1)}$$ for some $$c_k>0$$ with probability $$1-o(1)$$ at every given $$k\in\mathbb N$$.

However what is the minimum $$\gamma_k$$ in $$\max(A_1,\dots,A_k,B_1,\dots,B_k)-\min(A_1,\dots,A_k,B_1,\dots,B_k)\leq n^{\gamma_k}$$ to get typical value of $$\mu_k((A_1,B_1)\otimes(A_2,B_2)\otimes\dots\otimes(A_k,B_k))$$ to be at least $$\Omega(n^{k/(2^k-1)})$$ with probability $$1-o(1)$$ at every given $$k\in\mathbb N$$? Is $$n^{\gamma_k}$$ as low as $$n^{k/(2^k-1)}$$ or does at least $$2^k\gamma_k=2^{o(k)}$$ hold (at $$k=1$$ we have $$\gamma_1\rightarrow0$$)?

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## This question has an open bounty worth +50 reputation from Freeman. ending in 5 days.

The question is widely applicable to a large audience. A detailed canonical answer is required to address all the concerns.

• You delete more than half your questions. How about keeping this one open for a change. – Todd Trimble Oct 9 at 23:57
• Ok I shall do that. – Freeman. Oct 10 at 0:47