Potential estimates for a class of volume circuits with near outputs

In this paper,volume circuits are considered, which are the embeddings of circuits of functional elements in space. The class $ T_{\mathrm{near}} $ of circuits where the outputs are located side by side was considered. For this class, the lower and upper estimates of the potential are obtained. Potential is a measure of power equal to the number of circuit elements that produce one on a given input. In particular, it is shown that for Boolean operators with $ n $ inputs and $ m $ outputs, the order of the Shannon function for the $ T_{\mathrm{near}} $ circuit class is $ \Theta\left(\frac{m}{n} \cdot {\min}^{1/3}(m, 2^{n/2}) \cdot 2^{n/3} \right) $ for $ m \ge n $, $ \log_2(m) = o(2^n)$, $n \rightarrow \infty $.