We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve E t : Y 2 = X ( X 2 + ( t 2 4 - 2 ) X + 1 ) . . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of E t ( Q ¯ ( t ) ) is 0 and the torsion subgroup of E t ( Q ( t ) ) is isomorphic to Z ∕ 4 Z . Over the rational field Q we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of E t ( Q ) and conclude with some results regarding integral points in arithmetic progression on Et .
A set of positive integers $A \subset \mathbb{Z}_{>0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we study arithmetic progressions in sums of log-sparse subsets of $\mathbb{Z}_{>0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$ of $\mathbb{Z}_{>0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that this is nearly tight by proving that there exist log-sparse sets $S_1, \dots, S_n$ such that $S_1 + \cdots + S_n$ contains an arithmetic progression of size $n^{(1-o(1)) n}.$
