25 research outputs found
Quantitative bounds for the -inverse theorem over low characteristic finite fields
This paper gives the first quantitative bounds for the inverse theorem for
the Gowers -norm over when . We build upon earlier
work of Gowers and Mili\'cevi\'c who solved the corresponding problem for
. Our proof has two main steps: symmetrization and integration of
low-characteristic trilinear forms. We are able to solve the integration
problem for all -linear forms, but the symmetrization problem we are only
able to solve for trilinear forms. We pose several open problems about
symmetrization of low-characteristic -linear forms whose resolution,
combined with recent work of Gowers and Mili\'cevi\'c, would give quantitative
bounds for the inverse theorem for the Gowers -norm over
for all .Comment: 17 page
Hypergraph expanders of all uniformities from Cayley graphs
Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion
properties. In a recent paper, the first author gave a simple construction,
which can be randomized, of -uniform hypergraph expanders with
polylogarithmic degree. We generalize this construction, giving a simple
construction of -uniform hypergraph expanders for all .Comment: 32 page
Bounding sequence extremal functions with formations
An -formation is a concatenation of permutations of letters.
If is a sequence with distinct letters, then let be
the maximum length of any -sparse sequence with distinct letters which
has no subsequence isomorphic to . For every sequence define
, the formation width of , to be the minimum for which
there exists such that there is a subsequence isomorphic to in every
-formation. We use to prove upper bounds on
for sequences such that contains an alternation
with the same formation width as .
We generalize Nivasch's bounds on by showing that
and for every and , such that denotes the inverse Ackermann function.
Upper bounds on have been used in other
papers to bound the maximum number of edges in -quasiplanar graphs on
vertices with no pair of edges intersecting in more than points.
If is any sequence of the form such that is a letter,
is a nonempty sequence excluding with no repeated letters and is
obtained from by only moving the first letter of to another place in
, then we show that and . Furthermore we prove that
and for every .Comment: 25 page
Uniform sets with few progressions via colorings
Ruzsa asked whether there exist Fourier-uniform subsets of with density and 4-term arithmetic progression (4-APs)
density at most , for arbitrarily large . Gowers constructed
Fourier uniform sets with density and 4-AP density at most
for some small constant . We show that an affirmative
answer to Ruzsa's question would follow from the existence of an
-coloring of without symmetrically colored 4-APs. For a broad
and natural class of constructions of Fourier-uniform subsets of , we show that Ruzsa's question is equivalent to our arithmetic
Ramsey question.
We prove analogous results for all even-length APs. For each odd ,
we show that there exist -uniform subsets of
with density and -AP density at most .
We also prove generalizations to arbitrary one-dimensional patterns.Comment: 20 page
Triangle Ramsey numbers of complete graphs
A graph is -Ramsey if every two-coloring of its edges contains a
monochromatic copy of . Define the -Ramsey number of , denoted by
, to be the minimum number of copies of in a graph which is
-Ramsey. This generalizes the Ramsey number and size Ramsey number of a
graph. Addressing a question of Spiro, we prove that
for all sufficiently large . We do so
through a result on graph coloring: there exists an absolute constant such
that every -chromatic graph where every edge is contained in at least
triangles must contain at least triangles in total.Comment: 14 page
Induced arithmetic removal: complexity 1 patterns over finite fields
We prove an arithmetic analog of the induced graph removal lemma for
complexity 1 patterns over finite fields. Informally speaking, we show that
given a fixed collection of -colored complexity 1 arithmetic patterns over
, every coloring with density of every such pattern can be recolored on an
-fraction of the space so that no such pattern remains.Comment: 22 page