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The goal of this work on mathematical cosmology and geometric methods inmodified gravity theories, MGTs, is to investigate Starobinsky-like inflationscenarios determined by gravitational and scalar field configurations mimickingquasicrystal, QC, like structures. Such spacetime aperiodic QCs are differentfrom those discovered and studied in solid state physics but described bysimilar geometric methods. We prove that an inhomogeneous and locallyanisotropic gravitational and matter field effective QC mixed continuous anddiscrete "aether" can be modeled by exact cosmological solutions in MGTs andEinstein gravity. The coefficients of corresponding generic off-diagonalmetrics and generalized connections depend (in general) on all spacetimecoordinates via generating and integration functions and certain smooth anddiscrete parameters. Imposing additional nonholonomic constraints, prescribingsymmetries for generating functions and solving the boundary conditions forintegration functions and constants, we can model various nontrivial torsion QCstructures or extract cosmological Levi--Civita configurations with diagonalmetrics reproducing de Sitter (inflationary) like and other types homogeneousinflation and acceleration phases. Finally, we speculate how various darkenergy and dark matter effects can be modeled by off-diagonal interactions anddeformations of a nontrivial QC like gravitational vacuum structure andanalogous scalar matter fields.

Starobinsky Inflation and Dark Energy and Dark Matter Effects from Quasicrystal Like Spacetime Structures