230926 Jakub Byszewski

In relation to his work on Hilbert's third problem, Dehn proved in 1903 that a rectangle can be tiled by squares if and only if the ratio of its sides is a rational number. In 1994-95 Freiling-Rinne and Laczkovich-Szekeres solved a converse problem: for which real numbers r can one tile a square by rectangles with side ratio r. The answer is much more interesting: the property holds if and only if r is an algebraic number all of whose conjugates have positive real part. Soon afterwards the result was generalised to tilings of arbitrary rectangles by rectangles with a given side ratio. Unfortunately, in this case the corresponding criterion was somewhat mysterious, and it was not clear whether it was algorithmically decidable.

In the talk, we give a geometric interpretation of this condition and we show that it is decidable. Using these methods, we generalise the above results to tilings of squares by finitely many rectangles with possibly different side ratios under the extra assumption that all these ratios are algebraic numbers. We also classify all pairs of rectangles which are mutually tileable (there are nontrivial examples!). This is joint work with Radomił Baran.