  We provide a substantial improvement on a recent result by Dixit, Kim, and
Murty on the upper bound of $M_k(n)$, the largest size of a generalized
Diophantine tuple with property $D_k(n)$, that is, each pairwise product is $n$
less than a $k$-th power. In particular, we show $M_k(n)=o(\log n)$ for a
specially chosen sequence of $k$ and $n$ tending to infinity, breaking the
$\log n$ barrier unconditionally. One innovation of our proof is a novel
combination of Stepanov's method and Gallagher's larger sieve. One main
ingredient in our proof is a non-trivial upper bound on the maximum size of a
generalized Diophantine tuple over a finite field. In addition, we determine
the maximum size of an infinite family of generalized Diophantine tuples over
finite fields with square order, which is of independent interest.
  We also make significant progress towards a conjecture of S\'{a}rk\"{o}zy on
multiplicative decompositions of shifted multiplicative subgroups. In
particular, we prove that for almost all primes $p$, the set $\{x^2-1: x \in
\mathbb{F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two
sets in $\mathbb{F}_p$ non-trivially.
