Perfectly Packing an Equilateral Triangle by Equilateral Triangles of Sidelengths \(n^{-1/2-\epsilon }\)

Equilateral triangles of sidelengths 1, \(2^{-t}\) $2^{-t}$ , \(3^{-t}\) $3^{-t}$ , \(4^{-t},\ldots \ \) $4^{-t},\dots$ can be packed perfectly into an equilateral triangle, provided that \(\ 1/2<t \le 37/72\) $1/2<t\le 37/72$ . Moreover, for t slightly greater than 1/2, squares of sidelengths 1, \(2^{-t}\) $2^{-t}$ , \(3^{-t}\) $3^{-t}$ , \(4^{-t},\ldots \ \) $4^{-t},\dots$ can be packed perfectly into a square \(S_t\) $S_t$ in such a way that some squares have a side parallel to a diagonal of \(S_t\) $S_t$ and the remaining squares have a side parallel to a side of \(S_t\) $S_t$ .