Density-zero conjecture for local defect sets

Establish that the local defect set has relative density zero for every visibility datum whose branch-locus counting function has polynomial growth, whose local visibility conditions each possess a relative density, and whose large-prime tail satisfies a uniform estimate analogous to the paper’s large-prime-tail condition.

Background

The paper defines global visibility as the property that a branch point is the first integral point on its parameterized curve, and p-adic visibility as the corresponding absence of an earlier integral point with strictly smaller p-adic minimum valuation. The local defect set consists of points that are p-adically visible at every prime but are not globally visible.

For positive-weight homogeneous families, the authors prove exact local detectability: global visibility equals simultaneous p-adic visibility. For non-monomial polynomial families of the form y=qP(x), exact local detectability can fail, but the authors show that the defect set has density zero. These results motivate the conjecture that density-zero defects persist under the stated polynomial-growth, local-density, and large-prime-tail hypotheses.

References

These observations naturally lead us to make the following conjecture.

Let be a visibility datum such that #(;X) has polynomial growth, every local visibility condition has a relative density, and the corresponding large-prime tail satisfies a uniform estimate analogous to eq:large-prime-tail. Then $\mathfrak{d}^{bad}()=0$.

eq:large-prime-tail:

$\lim_{Y\to\infty}\limsup_{X\to\infty}\frac{|\mathcal{W}_Y(X)|}{|(;X)|}=0.$

— Local-global principles for visibility of lattice points on parameterized curves (2609.12177 - Chaubey et al., 10 Sep 2026) in Section ‘Open questions’, first Conjecture

The examples in Section \ref{sec:sparse} suggest the following conjecture.

Let n\geq 2, and let be a visibility datum whose branches are defined for all t>0 and have the form

\phi_\alpha(t)

\bigl(a_1(\alpha)t^{w_1},\dots, a_n(\alpha)t^{w_n}\bigr),

where w_1,\dots,w_n\in_{\geq 1} and a_i(\alpha)\in_{>0}. Assume that $\gcd(w_1,\dots,w_n)=1$ and suppose that () is sparse. Then, for every prime p, the relative density \mathfrak d_p^{{\operatorname{vis}()} should exist, the Euler product of the local densities should converge, and the relative density of globally visible points should exist and satisfy

\mathfrak d^{{\operatorname{vis}()}

\prod_p \mathfrak d_p^{{\operatorname{vis}(),} where both the global density and the local densities are taken relative to ().

— Local-global principles for visibility of lattice points on parameterized curves (2609.12177 - Chaubey et al., 10 Sep 2026) in Section ‘Open questions’, second Conjecture