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?
 A: Probably Dirichlet's approximation theorem gives best possible answer for large $N$. We can only reduce the number of simultaneous approximations using symmetries of $N$-gon. For example for $8$-gon it is necessary to approximate only one number $1/\sqrt2$ among $8$ numbers $e^{2\pi ik/8}$ $(k=0,\ldots 7)$. 
But starting from arbitrary $N$-gon inscribed into the unit circle we can find $t\le R$ such that 
$$|tx_1-a_1|\le\frac1{R^{\frac1{2N}}},|ty_1-b_1|\le\frac1{R^{\frac1{2N}}},\ldots |tx_N-a_N|\le\frac1{R^{\frac1{2N}}},|ty_N-b_N|\le\frac1{R^{\frac1{2N}}},$$
where $(x_1,y_1), \ldots, (x_N,y_N)$ are coordinates of initial $N$-gon. It means that we can achieve $\epsilon_1=\epsilon_2\asymp t^{-1-\frac1{2N}}$ in the circle of radius $t$ for infinitely many $t$.
If we want a better exponent, we can do the following. For arbitrary $N$ we can take $x_1=1$, $y_1=0$ (no approximation needed). The rest part of vertices is symmetrical. So for odd $N$ we need to approximate $(N-1)/2$ points ($N-1$ coordinates), for $N\equiv2\pmod4$ $(N-2)/4$ points ($(N-2)/2$ coordinates), for $N\equiv4\pmod8$ $(N-4)/8$ points, for $N\equiv0\pmod8$ $(N-2)/4$ points $1+(N-8)/8$ coordinates.
Also there is an algorithm (based on discrete Fourier transform) of complexity $O(t^2)$ which allows to find all best possible solutions in the circle of radius $t$. This algorithm was published in Algebra and number theory for mathematical schools. Collected problems (Russian, see page 138). 
It is very simple. We need so specify only one vertex to define our regular $N$-gon. So we take all points $A_1=(x_1,y_1)$ such that $0\le y_1\le x_1\le t$. For each $A_1$ we define $A_2',\ldots,A_N'$ rotating $A_1$ around the origin, and $A_2,\ldots,A_N$ as nearest integer points to $A_2',\ldots,A_N'$. After that it necessary to check the quality of resulting $N$-gon. We can say that $A_1,\ldots,A_N$ are $N$ complex numbers and calculate discrete Fourier series. First Fourier coefficient gives main cycle ($|C_1|\approx |OA_1|$) and the rest part of Fourier series forms an error term.
