2026/07/26 by Zihang Fang
#math.NT
For c>1 and an integer radix b≥2, we study the positive integers m for which mbk≤ cm<(m+1)bk for some k≥0; for integer c, this is the self-prefix leading-digit condition. We derive an exact shrinking-target criterion; for c≥2, an exact signed-discrepancy identity isolates both infinitude and the conjectural logarithmic count. For c≥2 with nonintegral logarithmic slope, Lambert W-1 inversion produces a candidate sequence with an eventual two-gap law and an exact counting formula; for (c,b)=(2,10) all consecutive candidate gaps are 3 or 4. For algebraic c with irrational logb c, the Lambert-root phases satisfy deterministic moving-target asymptotics in an explicit nontrivial power range strictly below the critical scale. For irrational logarithmic slope, actual hits obey fixed-difference and arithmetic-chain rigidity; for multiplicatively independent integer parameters, coherent endpoint hits at floor resonance centers force every intermediate term. Finally, set ρ=\logb c\. For fixed multiplicatively independent integers c,b, an interpolated continued-fraction locator has bit complexity O(N1-1/νpolylogN) for every ν>μ(ρ). We give an explicit certified instance for (2,10), whose infinitude remains open.