Convergence Results for Systems of Linear Forms on Cyclic Groups and Periodic Nilsequences

Given a positive integer $N$ and real number $\alpha\in [0, 1]$, let $m(\alpha,N)$ denote the minimum, over all sets $A\subseteq \mathbb{Z}_{N}$ of size at least $\alpha N$, of the normalized count of 3-term arithmetic progressions contained in $A$. A theorem of Croot states that $m(\alpha,N)$ converges as $N\to\infty$ through the primes, answering a question of Green. Using recent advances in higher-order Fourier analysis, we prove an extension of this theorem, showing that the result holds for $k$-term progressions for general $k$ and further for all systems of integer linear forms of finite complexity. We also obtain a similar convergence result for the maximum densities of sets free of solutions to systems of linear equations. These results rely on a regularity method for functions on finite cyclic groups that we frame in terms of periodic nilsequences, using in particular some regularity results of Szegedy (relying on his joint work with Camarena) and the equidistribution results of Green and Tao.

[1]  W. T. Gowers,et al.  A NEW PROOF OF SZEMER ´ EDI'S THEOREM , 2001 .

[2]  On higher order Fourier analysis , 2012, 1203.2260.

[3]  A. Leibman.  Polynomial Sequences in Groups , 1998 .

[4]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[5]  Ben Green,et al.  Sum-free sets in abelian groups , 2003 .

[6]  Bryna Kra,et al.  Nil–Bohr sets of integers , 2009, Ergodic Theory and Dynamical Systems.

[7]  P. Varnavides,et al.  On Certain Sets of Positive Density , 1959 .

[8]  Ernie Croot.  The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit , 2004, Canadian mathematical bulletin.

[9]  Terence Tao,et al.  The dichotomy between structure and randomness, arithmetic progressions, and the primes , 2005, math/0512114.

[10]  Jean Bourgain,et al.  On Triples in Arithmetic Progression , 1999 .

[11]  T. Schoen.  Linear Equations in Zp , 2005 .

[12]  Imre Z. Ruzsa,et al.  Solving a linear equation in a set of integers I , 1993 .

[13]  Olof Sisask,et al.  On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime , 2011, 1109.2934.

[14]  Terence Tao,et al.  An inverse theorem for the Gowers U^{s+1}[N]-norm , 2010 .

[15]  Daniel Král,et al.  On the removal lemma for linear systems over Abelian groups , 2013, Eur. J. Comb..

[16]  V. G. Mkhitaryan,et al.  On a class of homogeneous spaces of compact Lie groups , 1981 .

[17]  Ben Green,et al.  Linear equations in primes , 2006, math/0606088.

[18]  G. Bellamy.  Lie groups, Lie algebras, and their representations , 2015 .

[19]  Ben Green,et al.  The quantitative behaviour of polynomial orbits on nilmanifolds , 2007, 0709.3562.

[20]  Terence Tao,et al.  Higher Order Fourier Analysis , 2012 .

[21]  Ben Green,et al.  An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications , 2010, 1002.2028.

[22]  O. A. Camarena,et al.  Nilspaces, nilmanifolds and their morphisms , 2010, 1009.3825.

[23]  K. F. Roth.  On Certain Sets of Integers , 1953 .

[24]  A. Leibman.  Polynomial mappings of groups , 2002 .

[25]  Bryna Kra,et al.  Nonconventional ergodic averages and nilmanifolds , 2005 .

[26]  A. Leibman.  Orbit of the diagonal in the power of a nilmanifold , 2009 .

[27]  W. T. Gowers,et al.  The true complexity of a system of linear equations , 2007, 0711.0185.