DIMENSIONS OF SETS AVOIDING APPROXIMATE NONTRIVIAL ZEROS OF LINEAR PATTERNS

Abstract Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form $$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$ where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$ , $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$ , we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$ .