Sets of large dimension not containing polynomial configurations

Abstract

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Large sets avoiding patterns

“…It applies to a countable family of functions f : R nv → R with a fixed v that are not necessarily polynomials with rational coefficients. Further, in contrast with [14], the Hausdorff dimension of the obtained set depends on the number of vector variables v. The second result is of a perturbative flavour, and gives a set of positive Hausdorff dimension that simultaneously avoids zeros of all functions with a common linearization and bounded higher-order terms. To the best of our knowledge, such uniform avoidance results are new.…”

“…, p n (t)). Let us observe that the result in [14] does not apply to the non-polynomial function f 2 (t 1 , t 2 , t 3 ), but does apply to…”

“…has no solutions for any a ∈ A where x 1 , x 2 and x 3 are distinct points in E. Maga [13] exploits this idea to demonstrate a full-dimensional subset E ⊂ R n not containing the vertices of any parallelogram. Other results in this direction of considerable generality, extending their predecessors in [10,11,13], are due to Máthé [14]. Given any countable collection of polynomials p j : R nmj → R of degree at most d with rational coefficients, the main result of [14] ensures the existence of a subset E ⊆ R n of Hausdorff dimension n d such that p j (x 1 , .…”

“…, x mj ∈ E. The same conclusion continues to hold if the polynomials p j are replaced by p j (Φ j,1 (x 1 ), · · · , Φ j,mj (x mj )), where Φ j,k are C 1 -diffeomorphisms of R n . Interestingly, the Hausdorff dimension bound in [14], while depending on the ambient dimension n and the maximum degree d of the polynomials, is independent of the number of input vectors m j in p j , which may continue to grow without bound. This paper uses similar ideas to present two results in a somewhat different direction.…”

“…(4) Another point worth noting is that for the quadratic polynomial f (x 1 , x 2 , x 3 ) = (x 3 − x 1 ) − (x 2 − x 1 ) 2 , the set in R avoiding zeros of f is guaranteed to be of Hausdorff dimension 1 2 , both according to [14] and Theorem 1.1. It is not known however whether this bound is optimal.…”