In every dimension $d\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set $\mathcal C\subset \mathbb R^d$ of volume at least $v_d$ and $\mathcal U_d$ contains at most $\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$.
Deep Dive into Universal convex coverings.
In every dimension $d\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set $\mathcal C\subset \mathbb R^d$ of volume at least $v_d$ and $\mathcal U_d$ contains at most $\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$.
Fix a dimension d ≥ 1 and consider the volume associated to the standard Lebesgue measure on R d . Given v > 0, a subset U of R d is a v-universal convex covering of R d if we have C + U = R d for every convex subset C of R d of volume strictly greater than v. Here, C + U denotes the set of all points of the form P + Q with P in C and Q in U .
For every positive t, a subset U of R d is a v-universal convex covering if and only if tU is a (t d v)-universal convex covering. The properties in which we are interested are thus independent of the particular value of v. We call U a universal convex covering of R d if U is a v-universal convex covering of R d for some v > 0.
Our main result is the following. Theorem 1.1. Let d ≥ 1. Up to translation and rescaling, any universal convex covering of the Euclidean vector space R d has at least ℓ d (r) = r d points at distance at most r from the origin. There exists a universal convex covering U d of R d with at most u d (r) = log(r) d-1 r d points at distance at most r from the origin, for every large r.
The first part of theorem 1.1 is obvious since, in any dimension d, one can consider the unit cube as the convex set C. In dimension d = 1, the second part is obvious too since I + Z = R for every interval I of length strictly greater than 1, hence one can choose U 1 = Z. However, in dimension d ≥ 2, there is a factor of log d-1 between the easy lower bound ℓ d and the upper bound u d .
Whilst it is surely possible to improve these results, I would be very surprised by the existence of universal coverings in dimension d achieving the lower bound ℓ d for d ≥ 2. Proposition 2.1 below suggests thus the following question. Call a subset S of R n uniformly discrete if there exists a neighbourhood O of the origin such that x -y ∈ O for every pair (x, y) of distinct elements in S.
As long as the answer to question 1.2 is unknown or if the answer is NO, the following question and its higher-dimensional generalisations is also interesting.
Question 1.3. Does the complement of a uniformly discrete subset S of the plane necessarily contain triangles of arbitrarily large area?
Universal convex coverings are related to sphere coverings, see [1] for an overview, or more generally to coverings of R d by translates of a fixed convex body. Rogers proved in [4] that every convex body of R d covers R d with density at most d(5 + log d + log log d) for a suitable covering. Erdös and Rogers in [2] showed the existence of such a covering which furthermore covers no point with multiplicity exceeding ed(5 + log d + log log d). Chapter 31 of [3] contains an account of subsequent developpements.
There appears to be no result in the literature closely related to universal convex coverings and featuring results similar to theorem 1.1.
This paper is organized as follows. In section 2 we collect some preliminary facts. In section 3 we construct recursively a sequence of sets (U d ) d≥1 such that U 1 = Z and U d ⊂ R d for every d ≥ 1, and we show, by induction on d ≥ 1, that U d is a universal convex covering of R d . In section 4, we define growth classes of functions and we introduce a natural equivalence relation on them, which is compatible with the natural partial order on increasing positive functions. Finally, we show in section 5 that the growth class of the universal convex covering U d constructed in section 3 is represented by u d . This implies theorem 1.1 by rescaling U d suitably.
For any subset S of R d , let
denote the set of all opposite vectors.
The growth function f S of a subset S ⊂ R d without accumulation points is defined as follows: for r an arbitrary positive real number, f S (r) denotes the number of points of S at distance at most r from the origin.
Universal convex coverings are stable under affine bijections and v-universal convex coverings are stable under affine bijections which preserve the volume. Thus we consider the growth class with respect to the equivalence relation ∼ defined as follows: for any increasing non-negative functions f and g, we have f ∼ g if there exists a real number t ≥ 1 such that f (r) ≤ g(tr) ≤ f (t 2 r) for every r ≥ t.
Growth functions of sets without accumulation points related by affine bijections are equivalent under this equivalence relation.
For any nonzero integer n, let v 2 (n) denote the 2-valuation of n: this is the unique integer k such that n is 2 k times an odd integer. Write any point x of R d as x = (x i ) 1≤i≤d , use the coordinate functions π i defined by π i (x) = x i and let ρ (i) denote the projection of R d onto R d-1 obtained by erasing the i-th coordinate x i and defined by
3 From dimension d to dimension d + 1
Let U denote a subset of R d . For every 1
For example, ϕ 1 (Z) ⊂ R 2 is the set of all points (x, y) ∈ (Z[ 1 2 ]) 2 such that xy ∈ Z{0} or xy = 0 and x+y ∈ Z{0}. Otherwise stated, a point (x, y) of ϕ 1 (Z) is either a non-zero element of Z 2 or it has two non-zero coordinates and belongs to t
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