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We introduce the arithmetic width of a convex body, defined as the number of distinct values a linear functional attains on its lattice points. This notion refines lattice width by detecting gaps in the distribution of lattice points, while always yielding a natural lower bound. Our first result generalizes the structure theorem for sets of length from factorization theory: for large dilations of a convex body, the attained values form an arithmetic progression with only finitely many omissions near the extremes. In the spirit of Ehrhart theory, we further show that for rational polytopes, arithmetic width grows quasilinearly in the dilation parameter, with optimal directions recurring periodically. Finally, we discuss algorithms for computing arithmetic width. These results build new bridges among discrete geometry, integer programming, and additive combinatorics.