A note on uniform density in weak arithmetical theories

Answering a question raised by Shavrukov and Visser, we show that the lattice of $existsSigma^b_1$-sentences (in the language of Buss' weak arithmetical system $S^1_2$) over any computable enumerable consistent extension $T$ of $S^1_2$ is uniformly dense (in the sense of Definition 2). We also show that for every $mathcal{C} in {Phi_n: nge 3} cup {Theta_n: n ge 2}$ (where $Phi$ and $Theta$ refer to the known hierarchies of arithmetical formulas introduced by Burr for intuitionistic arithmetic) the lattices of $mathcal{C}$-sentences over any c.e. consistent extension $T$ of the intuitionistic version of Robinson Arithmetic $R$ are uniformly dense. As an immediate consequence of the proof, all these lattices are also locally universal (in the sense of Definition 3).