Rachel Greenfeld and I simply uploaded our article on arXiv”An example against the periodic tiling assumption“. That is the total model of the consequence I introduced on this weblog just a few months in the past. *periodic tiling assumption* associated to Grunbaum-Shephard And Lagarias-Wang. The article took a bit longer than anticipated as a consequence of a technical problem that we weren’t conscious of on the time of the announcement and wanted a workaround.

In additional element: The unique technique, as described within the announcement, was to create a “tiling language” that might encode a specific “sequence”.-adic Sudoku puzzle ”after which present that the second kind of puzzle has non-periodic options provided that: was a big sufficient prime quantity. It appears that evidently the second half of this technique labored, however there was an issue with the primary half: Our tile language (utilizing this) -group valued capabilities) to encode arbitrary boolean relationships between boolean capabilities and in addition (utilizing) -value capabilities) to encode “clock” capabilities akin to it was part of us -adic Sudoku puzzle, however we have not been in a position to get these two sorts of capabilities to “discuss” to one another as wanted to code. -adic Sudoku puzzle (primary downside, if a finite commutative -group then there aren’t any non-trivial subgroups. not included in or insignificant course). Consequently, we needed to change our “”.-adic Sudoku puzzle” by “-adic Sudoku puzzle” mainly means changing the prime by a sufficiently giant power (we imagine will suffice). This fastened the encoding problem, however -adic Sudoku puzzles had been a bit extra complicated than others. -adic situation, because of the following motive. Beneath is a pleasant evaluation train:

Theorem 1 (Linearity in three instructions means full linearity)To offer permission be an everyday perform that’s affine-linear on each horizontal line, diagonal (slope line) ) and anti-diagonal (slope line ). In different phrases, for any capabilities , And each affine perform . Later is an affine perform on .

Certainly, the property of being affine in three instructions signifies that there’s the second-order type related to Hessian. disappears at any level , And and so it ought to disappear all over the place. In actual fact, the smoothness speculation isn’t essential; We go away this as an train to the reader. If one is substituted, the identical expression seems to be true. with cyclic group if unusual; that is the important thing for us to indicate -adic Sudoku puzzles have a (roughly) two-dimensional affinity construction, which can be utilized later in additional evaluation to indicate that they’re truly non-periodic. Nonetheless, the corresponding declare for cyclic teams when can it fail is powerful sufficient ! Really the overall type of capabilities takes the affine type on every horizontal line, diagonal, and diagonal

for some integer coefficients . This extra time period “pseudo-affine” causes some extra technical issues, however finally seems to be manageable.

Throughout the writing course of, we additionally found that the coding a part of the proof turns into extra modular and conceptual when two new definitions, “expressible property” and “poorly expressible property”, are added. These ideas are considerably just like: sentences and sentences in arithmetic hierarchyor algebraic sets And semi-algebraic sets in actual algebraic geometry. Roughly talking, an expressible property is a property of a set of capabilities. , from a commutative group finite abelian teams such that the characteristic may be expressed on the graph by way of a number of tiling equations

For instance, the property that two capabilities have the distinction with respect to a continuing may be expressed by way of the tiling equation

(vertical line check) and in addition

The place is the diagonal subgroup . A poorly expressible characteristic is an existential measurement of some expressible trait so a set of capabilities suits property if and provided that there’s an extension of this tuple by some extra performance that matches the characteristic . It seems that weak expressible options are turned off below a collection of helpful operations, permitting us to simply create fairly complicated weakly expressible options from a “library” of easy weakly expressible options, akin to a fancy pc program may be created. easy library routines. Specifically, we can “program” our Sudoku puzzle as a weakly expressive characteristic.

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