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2026
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| Online Access: | https://doi.org/10.5281/zenodo.20311511 |
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| _version_ | 1866902136690835456 |
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| author | Su, Zhen |
| author_facet | Su, Zhen |
| contents | <p>This paper develops a first-principle conservation framework for lattice point discrepancies, relying only on set theory, Euclidean geometry, and symmetry, without advanced analytic number theory tools such as the circle method or exponential sums. We introduce a radial oscillatory flux to describe the discrepancy between discrete lattice counting and continuous area, decompose the flux into fast and slow oscillatory components, and rigorously prove the optimal error exponent for the Gauss circle problem. The fast oscillation cancels exactly due to modular-4 antisymmetry, while the slow component decays via conservation and contradiction arguments, yielding \(E(r)=O(r^{1/2})\). A complementary lower bound \(E(r)=\Omega(r^{1/2})\) is established by constructing a sequence of radii with constructive lattice interference. We further couple the flux to the sum-of-two-squares function \(r_2(n)\), derive its oscillatory structure, and give an elementary proof of Lagrange’s four-square theorem within the same framework. The theory is generalized to \(d\)-dimensional lattice sphere problems, with optimal exponents \(\theta_d=(d-1)/2\). The results are consistent with classical work and recent progress, offering a geometrically transparent, axiomatic alternative to analytic methods.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20311511 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Lattice Discrepancy Conservation and a First-Principle Proof of the Optimal Exponent for the Gauss Circle Problem Su, Zhen <p>This paper develops a first-principle conservation framework for lattice point discrepancies, relying only on set theory, Euclidean geometry, and symmetry, without advanced analytic number theory tools such as the circle method or exponential sums. We introduce a radial oscillatory flux to describe the discrepancy between discrete lattice counting and continuous area, decompose the flux into fast and slow oscillatory components, and rigorously prove the optimal error exponent for the Gauss circle problem. The fast oscillation cancels exactly due to modular-4 antisymmetry, while the slow component decays via conservation and contradiction arguments, yielding \(E(r)=O(r^{1/2})\). A complementary lower bound \(E(r)=\Omega(r^{1/2})\) is established by constructing a sequence of radii with constructive lattice interference. We further couple the flux to the sum-of-two-squares function \(r_2(n)\), derive its oscillatory structure, and give an elementary proof of Lagrange’s four-square theorem within the same framework. The theory is generalized to \(d\)-dimensional lattice sphere problems, with optimal exponents \(\theta_d=(d-1)/2\). The results are consistent with classical work and recent progress, offering a geometrically transparent, axiomatic alternative to analytic methods.</p> |
| title | Lattice Discrepancy Conservation and a First-Principle Proof of the Optimal Exponent for the Gauss Circle Problem |
| url | https://doi.org/10.5281/zenodo.20311511 |