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Igor Rivin
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Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within $\epsilon_1$ of $1,$ and the ratio of the largest to the smallest angle is within $\epsilon_2$ of $1$) The question is: is there any sort of reasonable upper/lower bound to the size (area or diameter) of such a thing? This is clearly related to diophantine approximation in the following way: take the "standard" regular $N$-gon (one whose vertices are $N$-th roots of unity,) and blow it up by a factor of $t$ until wall the vertices ($t$ times the roots of unity) are close to lattice points. How big does $t$ need to be in terms of some "closeness" bound?

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within $\epsilon_1$ of $1,$ and the ratio of the largest to the smallest angle is within $\epsilon_2$ of $1$) The question is: is there any sort of reasonable upper/lower bound to the size (area or diameter) of such a thing? This is clearly related to diophantine approximation in the following way: take the "standard" regular $N$-gon (one whose vertices are $N$-th roots of unity, and blow it up by a factor of $t$ until wall the vertices ($t$ times the roots of unity) are close to lattice points. How big does $t$ need to be in terms of some "closeness" bound?

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within $\epsilon_1$ of $1,$ and the ratio of the largest to the smallest angle is within $\epsilon_2$ of $1$) The question is: is there any sort of reasonable upper/lower bound to the size (area or diameter) of such a thing? This is clearly related to diophantine approximation in the following way: take the "standard" regular $N$-gon (one whose vertices are $N$-th roots of unity,) and blow it up by a factor of $t$ until wall the vertices ($t$ times the roots of unity) are close to lattice points. How big does $t$ need to be in terms of some "closeness" bound?

Source Link
Igor Rivin
  • 96.4k
  • 11
  • 153
  • 366

Regular lattice polygons

Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within $\epsilon_1$ of $1,$ and the ratio of the largest to the smallest angle is within $\epsilon_2$ of $1$) The question is: is there any sort of reasonable upper/lower bound to the size (area or diameter) of such a thing? This is clearly related to diophantine approximation in the following way: take the "standard" regular $N$-gon (one whose vertices are $N$-th roots of unity, and blow it up by a factor of $t$ until wall the vertices ($t$ times the roots of unity) are close to lattice points. How big does $t$ need to be in terms of some "closeness" bound?