Spectral sets in Zp2qr

We prove the every spectral set in Zp2qr tiles, where p, q and r are primes. Combining this with a recent result of Malikiosis we obtain that Fuglede's conjecture holds for Zp2qr

Non-expander Cayley graphs of simple groups

For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders

Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the possible decompositions of the transcendental extension of the rational field of rank one into two pieces. Further, we prove that the positive elements of a real algebraic extensions of the rational numbers are indecomposable into two pieces

Minimum area isosceles containers

We show that every minimum area isosceles triangle containing a given triangle $T$ shares a side and an angle with $T$. This proves a conjecture of Nandakumar motivated by a computational problem. We use our result to deduce that for every triangle $T$, (1) there are at most $3$ minimum area isosceles triangles that contain $T$, and (2) there exists an isosceles triangle containing $T$ whose area is smaller than $\sqrt2$ times the area of $T$. Both bounds are best possible