The Hasse principle for random homogeneous polynomials in thin sets

Let $d$ and $n$ be natural numbers. Let ${\nu }_{d,n}:{\mathbb{R}}^n\to {\mathbb{R}}^N$ denote the Veronese embedding with $N=N_{n,d}:=\left(\genfrac{}{}{0px}{}{n+d-1}{d}\right)$, defined by listing all the monomials of degree $d$ in $n$ variables using the lexicographical ordering. Let $⟨\mathit{a},{\nu }_{d,n}(\mathit{x})⟩\in \mathbb{Z}[\mathit{x}]$ be a homogeneous polynomial in $n$ variables of degree $d$ with integer coefficients $\mathit{a}$, where $⟨\cdot ,\cdot ⟩$ denotes the inner product. For a non-singular form $P\in \mathbb{Z}[\mathit{x}]$ of degree $k(\le d)$ in $N$ variables, consider a set of integer vectors $\mathit{a}\in {\mathbb{Z}}^N$, defined by $$\mathfrak{A}(A;P)=\{\mathit{a}\in {\mathbb{Z}}^N:P(\mathit{a})=0,\mathrm{\parallel }\mathit{a}{\mathrm{\parallel }}_{\mathrm{\infty }}\le A\}\mathrm{.}$$ By handling a new lattice problem via the geometry of numbers, we confirm that whenever $n>24d$ and $d\ge 17,$ the proportion of integer coefficients $\mathit{a}\in \mathfrak{A}(A;P)$, whose associated equation $f_{\mathit{a}}(\mathit{x})=0$ satisfies the Hasse principle, converges to $1$ as $A\to \mathrm{\infty }$. This improves on the recent work of the second author.