We consider estimation in a sparse additive regression model with the design points on a regular lattice. We establish the minimax convergence rates over Sobolev classes and propose a Fourier-based rate-optimal estimator which is adaptive to the unknown sparsity and smoothness of the response function. The estimator is derived within Bayesian formalism but can be naturally viewed as a penalized maximum likelihood estimator with the complexity penalties on the number of nonzero univariate additive components of the response and on the numbers of the nonzero coefficients of their Fourer expansions. We compare it with several existing counterparts and perform a short simulation study to demonstrate its performance.