Possible Sizes of Sumsets

Isaac Rajagopal

doi:10.19086/da.165102

combinatorial number theory

Possible Sizes of Sumsets

arXiv:2510.23022 · doi:10.19086/da.165102

summary

The paper determines the possible cardinalities of h‑fold sumsets of large subsets of integers, showing that for sufficiently large set size k the range of sizes is an explicit interval missing only a small, specified set of values.

Abstract

Nathanson introduced the range of cardinalities of $h$ -fold sumsets $R (h, k) := {∣ h A ∣ : A \subset Z and ∣ A ∣ = k} .$ Following a remark of ErdÅs and SzemerÃ©di that determined the form of $R (h, k)$ when $h = 2$ , Nathanson asked what the form of $R (h, k)$ is for arbitrary $h, k \in N$ . For $h \in N$ , we prove there is some constant $k_{h} \in N$ such that if $k > k_{h}$ , then $R (h, k)$ is the entire interval $[hk - h + 1, (h h + k - 1)]$ except for a specified set of $(2 h - 1)$ numbers. Moreover, we show that one can take $k_{3} = 2$ .

17 pages, 4 figures, version published by Discrete Analysis

Topics & keywords

#sumsets #additive combinatorics #h‑fold sumset #cardinality intervals #extremal combinatoricsR(h,k)h‑fold sumsetcardinalitybinomial coefficientErdős–SzemerédiNathanson