A basic theory of Benford’s Law

Abstract

Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford's Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.

Benford’s Distribution in Extrasolar World: Do the Exoplanets Follow Benford’s Distribution?

“…Since the pioneering works of Newcomb and Benford, a large number of works related to Benford's law have been reported in various contexts (for a fascinating history of Benford's law, see [3][4][5]). For example, it's presence and applicability have been investigated in various domains, like astrophysics [6,7], geography [8], biology [9][10][11], seismography [12], stock market and accounting [13,14].…”

A Characterization of Benford’s Law in Discrete-Time Linear Systems

“…Since log 10 2 n = n log 10 2 ∈ span Q {1, log 10 2} ⊂ span Q ∆ σ(A n ) , the set σ(A n ) is b-resonant for every n ∈ N. Correspondingly, there exist x, y ∈ R 6 for which the sequence (x ⊤ A n y), and in fact (|A n x|) as well, is neither 10-Benford nor terminating. Essentially the same calculation as in Example 3.14 shows that one can take for instance x = y = e (6) . Note, however, that (x ⊤ A n y) is 10-Benford whenever |x 1 y 2 | = |x 3 y 4 |, hence for most x, y ∈ R 6 ; see also Theorem 4.1 below.…”

“…Earlier, weaker forms and variants of the implication (ii)⇒(i) in Theorems 3.4 and 3.16, or special cases thereof, can be traced back at least to [32] and may also be found in [4,6,9,22,36]. The reverse implication (i)⇒(ii) seems to have been addressed previously only for d < 4; see [6,Thm.5.37]. For the special case of b = 10, partial proofs of Theorems 3.4 and 3.16 have been presented in [5,7].…”

“…For itself, this could be seen directly by noticing that the map Λ u : T → T has a non-degenerate critical point whenever βu 1 = 0, and hence cannot possibly preserve λ T , see e.g. [5,Lem.2.6] or [6,Ex.5.27(iii)]. The more elaborate calculation given here, however, is useful also in the second step of the proof, i.e.…”

“…In other words, most real d × d-matrices, both in a topological and measuretheoretical sense, belong to b∈N\{1} G d,b , and thus are invertible with their spectrum b-nonresonant for every b; see e.g. [4,8,6] for details. This observation may help explain the conformance to BL often observed empirically across a wide range of scientific disciplines.…”