Inhomogeneous Approximation by Sums of Roots

Abstract

Let $d\ge 2$ and $k\ge 1$ be fixed. We prove that, for every $\epsilon >0$ and every real $\beta$, there exist integers $1\le b_1,\dots ,b_k\le N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-β\right\| \ll_{d,k,ε} N^{-k/d+ε}. \] The proof combines Schmidt's Subspace Theorem with an explicit inhomogeneous transference argument. This improves Iyer's (2025) higher-root exponent $(k-d+1)/d^2$, and also the analogous $d$-ary full-basis exponent away from the cases where $k+1$ is a power of $d$, at the cost of ineffectivity. We also record a conjectural uniform exponent $k-1/d$. In the square-root case $d=2$, we give explicit integer-target constructions for $k=2,3,4$ attaining this conjectural value.