  We make quantitative improvements to recently obtained results on the
structure of the image of a large difference set under certain quadratic forms
and other homogeneous polynomials. Previous proofs used deep results of
Benoist-Quint on random walks in certain subgroups of
$\operatorname{SL}_r(\mathbb{Z})$ (the symmetry groups of these quadratic
forms) that were not of a quantitative nature. Our new observation relies on
noticing that rather than studying random walks, one can obtain more
quantitative results by considering polynomial orbits of these group actions
that are not contained in cosets of submodules of $\mathbb{Z}^r$ of small
index. Our main new technical tool is a uniform Furstenberg-S\'{a}rk\"{o}zy
theorem that holds for a large class of polynomials not necessarily vanishing
at zero, which may be of independent interest and is derived from a density
increment argument and Hua's bound on polynomial exponential sums.
