Let S be a square and let S' be a square of unit area with a diagonal parallel to a side of S. Any (finite or infinite) sequence of homothetic copies of S whose total area does not exceed 4/9 can be packed translatively into S'.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-2-9, author = {Janusz Januszewski}, title = {Translative packing of a square with sequences of squares}, journal = {Colloquium Mathematicae}, volume = {120}, year = {2010}, pages = {273-280}, zbl = {1214.52006}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-2-9} }
Janusz Januszewski. Translative packing of a square with sequences of squares. Colloquium Mathematicae, Tome 120 (2010) pp. 273-280. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-2-9/