Benford Behaviors in Stick and Box Fragmentation Models by Bruce Fang ’25

Benford’s law is the statement that in many real-world data set, the probability of having digit $d$, where $1leq dleq 9$, as the first digit is $log_{10}left((d+1)/dright)$. We sometimes refer to this as the weak Benford’s behavior, whereas we say that a data set exhibits the strong Benford’s behavior if the probability of having significant at most $s$, where $sin [1,10)$, is $log_{10}(s)$. We examine Benford behaviors in two different probabilistic models: stick fragmentation model and box fragmentation model. Building on the joint work of SMALL 11′ and 13′ on the 1-dimensional stick fragmentation model, we employ combinatorial identities on multinomial coefficients to reduce the high dimensional stick fragmentation model to the 1-dimensional model and provide a necessary and sufficient condition for the lengths of the stick to converge to strong Benford behavior. Then we answer a conjecture posed by SMALL 22′ on the high dimensional box fragmentation model. Using tools from Fourier analysis and order statistics, we prove that under some mild conditions, faces of any arbitrary dimension of the box have total volume converging to strong Benford behavior.