Arithmetic Progressions in Sparse Pseudorandom Sets via the Transference Principle in Additive Combinatorics

Mathematical Applications of Science Fiction We present a rigorous derivation of the existence of k-termarithmetic progressions in sparse subsets of the cyclic groupZN , provided the subset is majorized by a pseudorandommeasure ν satisfying the linear forms condition and thecorrelation condition. By invoking a hyper-local variantof the Green-Tao Transference Principle, we construct adense model g indistinguishable from the sparse functionf under the Gowers U k-norm topology. We extend theclassical Szemer´edi theorem to the sparse setting through adecomposition of the target function into structured, small,and uniform components, f = fstr + fsml + funf . Weprovide explicit error bounds for the counting operator Λkand demonstrate that for any set A ⊆ ZN with relativedensity δ > 0 with respect to a k-pseudorandom measure,the count of k-term arithmetic progressions is asymptoticallyδkN 2. The methodology integrates higher-order Fourieranalysis, Bohr set localizations, and transfinite inductionon the complexity of the linear systems involved.