2025/12/03 by Joseph Harrison, Harrison, Joseph
Mathematics · #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.2512.04081
20 pages, no figures, comments welcome!
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We investigate the interaction between raising to an irrational power and addition of real numbers. Thus, for a finite set A of non-negative real numbers, let A[c] = \ac : a ∈ A\. When k is a positive integer, c is a real irrational number, and A is a subset of an N-term arithmetic progression in ℝ≥ 0 having cardinality at least a power of logN, we prove that the k-fold sumset |kA[c]| ∼k |A|k/k! as |A| → ∞. This result is uniform in c. When A = \1, …, N\ and k = 2, this result can be combined with existing works to show that |A[c] + A[c]| ∼ N2/2 as N → ∞ whenever c ∈ ℝ ∖ \0, 1, 2\. The sumset lower bound follows from a bound on the number of equal sums of r and s ≥ r elements of A[c] (by taking r = s = k). When r = s = 2 or s > r, our bound is optimal up to a power of log N. This bound is proved using a functional transcendence theorem for certain endomorphisms of ℝ>0n, and innovations in the Pila--Wilkie counting theorem in ℝexp due to Binyamini, Novikov and Zak. In a different direction, we provide a Diophantine approximation criterion on c that, when satisfied, ensures that a linear form in the c-th powers of multiplicatively independent integers does not vanish. The proof involves linear forms in logarithms. This provides a new proof of a fact, due to Bays--Kirby--Wilkie and Jones--Servi, that when A is a multiplicatively independent set of positive integers, there are infinitely many effectively computable real numbers c such that A[c] is linearly independent over ℚ.