2019/09/18 by Brown, Erin Crossen, Mkrtchyan, Sevak, Pakianathan, Jonathan
#11P81 #60C05. Secondary: 11P70 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary: 05A17 #Probability (math.PR)
paper · doi:10.48550/arxiv.1909.08688
We study a process of generating random positive integer weight sequences \ Wn \ where the gaps between the weights \ Xn = Wn - Wn-1 \ are i.i.d. positive integer-valued random variables. We show that as long as the gap distribution has finite (1)/(2)-moment, almost surely, the resulting weight sequence is asymptotically complete, i.e., all large enough multiples of the gcd of the possible gap values can be written as a sum of distinct weights. We then show a much stronger result that if the gap distribution has a moment generating function with large enough radius of convergence, then every large enough multiple of the gcd of gap values can be written as a sum of m distinct weights for any fixed m ≥ 2.