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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2602.06891 |
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| _version_ | 1866917254657998848 |
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| author | Singh, Shalender Singh, Vishnupriya |
| author_facet | Singh, Shalender Singh, Vishnupriya |
| contents | We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure supported on annihilator submodules arising from the arithmetic of $n$.
As a consequence, we obtain the first inverse theorem for the Falconer distance problem over $\mathbb{Z}_n$ for composite moduli. We show that if a set $E \subset \mathbb{Z}_n^d$ of size $|E| \asymp n^{(d+1)/2}$ determines only $O(n)$ distinct squared distances, then $E$ must be supported on a coset of an annihilator submodule on which the distance form is algebraically degenerate.
The proof introduces a divisor-depth decomposition intrinsic to $\mathbb{Z}_n$, together with a lifting mechanism that transfers local degeneracies at prime moduli into global ideal-theoretic constraints. This yields a complete classification of near-extremizers for the Falconer distance problem in the ring setting, revealing a rigidity phenomenon with no analogue over fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06891 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inverse Falconer Distance Theorems over the Integer Residue Rings $\mathbb{Z}_n$ Singh, Shalender Singh, Vishnupriya Number Theory Combinatorics We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure supported on annihilator submodules arising from the arithmetic of $n$. As a consequence, we obtain the first inverse theorem for the Falconer distance problem over $\mathbb{Z}_n$ for composite moduli. We show that if a set $E \subset \mathbb{Z}_n^d$ of size $|E| \asymp n^{(d+1)/2}$ determines only $O(n)$ distinct squared distances, then $E$ must be supported on a coset of an annihilator submodule on which the distance form is algebraically degenerate. The proof introduces a divisor-depth decomposition intrinsic to $\mathbb{Z}_n$, together with a lifting mechanism that transfers local degeneracies at prime moduli into global ideal-theoretic constraints. This yields a complete classification of near-extremizers for the Falconer distance problem in the ring setting, revealing a rigidity phenomenon with no analogue over fields. |
| title | Inverse Falconer Distance Theorems over the Integer Residue Rings $\mathbb{Z}_n$ |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2602.06891 |