Functions Tiling with Several Lattices

Functions tiling simultaneously with two arithmetic progressions

“…We note that the correspondence between the 𝑛 × 𝑚 doubly stochastic arrays and the nonnegative functions that tile the group ℤ 𝑛𝑚 by translations along each one of the two subgroups 𝐻 𝑛 and 𝐻 𝑚 (where 𝑛, 𝑚 are coprime) was pointed out in an earlier version of [7]. Then g is supported on the set {0, 1, … , 𝑛 + 𝑚 − 2} of size 𝑛 + 𝑚 − 1.…”

“…Tilings by translates of a function have been studied by various authors, see, for example, [3][4][5][6][7][8][9][10][11][12].…”

“…It is obvious that if 𝑝, 𝑞 are both nonzero, then the simultaneous tiling condition (1.3) implies that mes(supp 𝑓) can be no smaller than max{𝛼, 𝛽}. This estimate was improved for nonnegative functions 𝑓 in [7,Theorem 2.6], where the authors proved that if 0 < 𝛼 < 𝛽 then the tiling condition (1.3) implies that mes(supp 𝑓) ⩾ ⌈𝛽∕𝛼⌉𝛼. The authors asked in [7, Question 4] what is the least possible measure of the support of a function 𝑓 satisfying (1.3).…”

“…This estimate was improved for nonnegative functions $$f$$ in [7, Theorem 2.6], where the authors proved that if $$0 &lt; \alpha &lt; \beta$$ then the tiling condition () implies that $$\operatorname{mes}(\operatorname{supp}f) \geqslant \lceil \beta /\alpha \rceil \alpha$$. The authors asked in [7, Question 4] what is the least possible measure of the support of a function $$f$$ satisfying (). In this paper, we obtain sharp results which improve on the lower bound from [7] and provide a complete answer to this question.…”

“…In [7], inspired by the Steinhaus tiling problem, the authors studied the following question: how “small” can be the support of a function $$f$$ that tiles $$\mathbb {R}^d$$ simultaneously by a finite number of lattices $$\Lambda _1, \dots , \Lambda _N$$? In particular, they posed the question as to what is the least possible measure of the support of such a function $$f$$.…”