2025/03/17 by Yuchen Ding, Ding, Yuchen
Mathematics · #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2503.22700
We study two additive problems which go back to Erdős' work around Romanoff's theorem and in which the sparse summand is a sequence of powers. In the prime case we prove a quantitative metric form of the Erdős--Kalmár problem: if Sy=\p+\lfloor yk rfloor:p\inP, k≥1\, δy=\liminfN→∞(|Sy∩[1,N]|)/(N), then for Lebesgue almost all y>1, δy≥ (1)/(log y+C ζ(2)/ζ(4)), where C is the absolute Selberg-sieve constant defined in (2.2). The dependence on y has the correct order as y→∞: a simple counting upper bound shows that no lower bound depending only on y can have order larger than 1/log y. We also prove a complementary exceptional-base statement: even in the real-base setting one cannot expect density-one coverage in general. For the golden ratio φ=(1+√5)/2, the set of integers not representable as p+\lfloorφk \rfloor, p\inP, k≥1, has positive lower density. The second problem is the square-free analogue of the power-of-two questions raised by Erdős in his 1950 paper on integers of the form 2k+p and related problems. Erdős conjectured, in particular, that every sufficiently large odd integer should be a square-free integer plus a power of two; this fixed-base problem remains open. We prove a variable-base density-one analogue: there exists a∈(2,3) such that #\n≤ x:n∉ Q+\\lfloor am\rfloor:m≥1\\=o(x), where Q denotes the positive square-free integers.