PRODUCT DISTRIBUTION IN A FIXED DEGREE CHAIN

Abstract

We study multiplicative structures arising from \emph{fixed-degree} addition chains: increasing sequences s₀=1<s₁<⋯<s_h in which each term is the sum of at most d≥ 2 earlier terms (repetition allowed). Writing 𝒫(L) for the set of integers that can be expressed as a product of distinct terms drawn from a (finite or infinite) fixed-degree chain L={T_m}, we develop a structural partition of chain steps into four classes 𝒜_d,ℬ_d,𝒞_d,𝒟_d (dilate / large / medium / small) and introduce the notion of a \emph{run} of a given step type. By deriving precise two-sided log-sum (product) estimates along pure and mixed-type runs and translating these into elementary counting bounds (for example, run-level bounds of the shape ≪(log x)^1/2 in the logarithmic scale), we obtain a global sparsity result: under the mild structural hypothesis that the number of runs of types 𝒟_d and 𝒜_d∪𝒟_d on initial segments of the chain is O_d(1), the product-set 𝒫(L) has zero asymptotic density. In particular, for any such infinite chain L one has lim_{x⟶ ∞}(#{v≤ x : v∈𝒫(L)})/x=0. The proof combines combinatorial run-analysis, elementary exponential growth estimates along runs, and an assembly of run-level contributions into explicit global counting bounds; quantitative run-level inequalities (involving the clearly defined parameters λ_d and a slowly varying δ(m)=1/log m) are recorded and may be of independent interest for further study of fixed-degree chains

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BibTeX RIS

BibTeX

@misc{agama2026product,
  title = {{PRODUCT DISTRIBUTION IN A FIXED DEGREE CHAIN}},
  author = {Theophilus Agama},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  url = {https://hal.science/hal-05784851v1},
  note = {CC BY-ND 4.0},
}

RIS

TY  - UNPB
TI  - PRODUCT DISTRIBUTION IN A FIXED DEGREE CHAIN
AU  - Theophilus Agama
PY  - 2026
DA  - 2026/10/07
PB  - HAL
UR  - https://hal.science/hal-05784851v1
AB  - We study multiplicative structures arising from \emph{fixed-degree} addition chains: increasing sequences s_0=1<s_1<⋯<s_h in which each term is the sum of at most d≥ 2 earlier terms (repetition allowed). Writing 𝒫(L) for the set of integers that can be expressed as a product of distinct terms drawn from a (finite or infinite) fixed-degree chain L={T_m}, we develop a structural partition of chain steps into four classes 𝒜_d,ℬ_d,𝒞_d,𝒟_d (dilate / large / medium / small) and introduce the notion of a \emph{run} of a given step type. By deriving precise two-sided log-sum (product) estimates along pure and mixed-type runs and translating these into elementary counting bounds (for example, run-level bounds of the shape ≪(log x)^1/2 in the logarithmic scale), we obtain a global sparsity result: under the mild structural hypothesis that the number of runs of types 𝒟_d and 𝒜_d∪𝒟_d on initial segments of the chain is O_d(1), the product-set 𝒫(L) has zero asymptotic density. In particular, for any such infinite chain L one has lim_x⟶ ∞(#{v≤ x : v∈𝒫(L)})/x=0. The proof combines combinatorial run-analysis, elementary exponential growth estimates along runs, and an assembly of run-level contributions into explicit global counting bounds; quantitative run-level inequalities (involving the clearly defined parameters λ_d and a slowly varying δ(m)=1/log m) are recorded and may be of independent interest for further study of fixed-degree chains
ER  -

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