Inger Johanne Håland Knutson

• Bergelson, Vitaly; Knutson, Inger J. Håland & Son, Younghwan (2021). An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications. International Mathematics Research Notices (IMRN). ISSN 1073-7928. 2021(19), p. 14965–15018. doi: 10.1093/imrn/rnaa035. Full text in Research Archive Show summary

  Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial q(n) has the property that the sequence (q(n)λ)n∈Z is well-distributed mod1 for all but countably many λ∈R if and only if lim|n|→∞n∉J|q(n)|=∞ for some (possibly empty) set J having zero natural density in Z⁠. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.

• Bergelson, Vitaly & Knutson, Inger Johanne Håland (2009). Weak mixing implies weak mixing of higher orders along tempered functions. Ergodic Theory and Dynamical Systems. ISSN 0143-3857. 29(5), p. 1375–1416. doi: 10.1017/S0143385708000862. Show summary

  We extend the weakly mixing PET (polynomial ergodic theorem) obtained in Bergelson [Weakly mixing PET. Ergod. Th. & Dynam. Sys. 7 (1987), 337- 349] to much wider families of functions. Besides throwing new light on the question of 'how much higher-degree mixing is hidden in weak mixing', the obtained results also show the way to possible new extensions of the polynomial Szemeredi theorem obtained in Bergelson and Leibman [Polynomial extensions of van der Waerden's and Szemeredi's theorems. J. Amer. Math. Soc. 9 (1996), 725-753].

• Bergelson, Vitaly; Knutson, Inger Johanne Håland & McCutcheon, Randall (2006). IP-systems, generalized polynomials and recurrence. Ergodic Theory and Dynamical Systems. ISSN 0143-3857. 26(4), p. 999–1019. Show summary

  Let Omega be an abelian group. A set R subset of 2 is a set of recurrence if for any probability measure-preserving system (X, B, mu, (T-g)(g is an element of Omega)) and any A is an element of A with mu(A) > 0, A(A boolean AND T(g)A) > 0 for some g is an element of R. If (x(i))(i=1)(infinity) is a sequence in Omega, the set of its finite sums {xi(1) + xi(2) + (...) + xi(k) : i(1) < (i)2 < (...) < i(k)} is called an IP-set. In Bergelson et al (Erg. Theor Dyn. Sys. 16 (1996), 963-974) it is shown that if p : Z(d) -> Z(k) is a polynomial vanishing at zero and F is an IP-set in Z(d) then {p(n) : n is an element of F} is a set of recurrence in Zk. Here we extend this result to an analogous family of generalized polynomials, that is functions formed from regular polynomials by iterated use of the greatest integer function, as a consequence of a theorem establishing a much wider class of recurrence sets occurring in any (possibly non-finitely generated) abelian group. While these sets do in a sense have a distinctively 'polynomial' nature, this far-ranging class includes, even in Z, such examples as {Sigma i,j is an element of alpha,i < j 2(i)3(j) : alpha subset of N, 0 < vertical bar alpha vertical bar < infinity}, where the connection to conventional polynomials is somewhat distant.

• Bergelson, Vitaly; Knutson, Inger Johanne Håland & McCutcheon, Randall (2005). Simultaneous diophantine approximation and VIP systems. Acta Arithmetica. ISSN 0065-1036. 116(1), p. 13–23.

View all works in Cristin

• Knutson, Inger J. Håland (2019). Minnetale over Anne Berit Fuglestad.
• Knutson, Inger J. Håland (2019). Examples of generalized polynomials and their dynamical properties/applications.
• Knutson, Inger J. Håland (2018). Matematikk i hemmelige koder.
• Knutson, Inger J. Håland (2018). Abelprisen 2017. In Jahr, Ernst Håkon; Nossum, Rolf Tomas; Thygesen, Ragnar & Breen, Olav (Ed.), Agder Vitenskapsakademi, Årbok 2017. Portal forlag. ISSN 978-82-02-59723-8. p. 254–256.
• Knutson, Inger J. Håland (2017). Abelprisen 2017.
• Knutson, Inger J. Håland (2016). Matematikk i hemmelige koder.
• Knutson, Inger J. Håland (2016). Experience of being a tutor at mathematics support centre at UiA . Show summary

  The support centre opened in August of 2015 and I have been a tutor there two hours a week since then. I will share some of my experiences and reflections around the enjoyable work in the support centre. In the fall I taught a mathematics course for economics students, and many of the students I tutored were taking this course. My reflections will be based on my fresh experiences in the centre but will also be influenced by my background as a teacher (professor) of a variety of mathematics courses over the years.

• Knutson, Inger J. Håland (2014). Akademiets kvinneforskermaraton 2013. Innledning. In Jahr, Ernst Håkon; Nossum, Rolf Tomas & Storesletten, Leiv (Ed.), Agder Vitenskapsakademi Årbok 2013. Portal forlag. ISSN 9788283140200. p. 166–167.
• Knutson, Inger J. Håland (2014). Smakebiter fra samspillet mellom ergodeteori og tallteori.
• Knutson, Inger J. Håland (2013). Abelprisen 2012:Endre Szemerédi. In Nossum, Rolf Tomas (Eds.), Agder vitenskapsakademi. Årbok 2012. Portal forlag. ISSN 978-82-92712-95-5. p. 364–367.
• Knutson, Inger J. Håland (2012). Abelprisen 2012 - Endre Szemerédi.
• Knutson, Inger J. Håland (2012). Årets Abelprisvinner, Endre Szemerédi.
• Knutson, Inger Johanne Håland (2008). How to generate polynomials dynamically. Show summary

  If we evaluate a polynomial along the natural numbers we get a sequence of numbers. For example, the sequence we get from the polynomial p(x)=ax is p(n)=an, n=1,2,... Such sequences can be generated by a dynamical system. In the talk I will explain what I mean by this and show how to find such a dynamical system. If time permits I will also show how to use matrices to generate certain "generalized polynomials".

• Knutson, Inger Johanne Håland (2008). An ergodic approach to problems in combinatorical number theory. Show summary

  Today there are a few proofs of Szemeredi's theorem (1974) on arithmetic progressions in "large" subsets of the integers. One of the proofs, due to Furstenberg, is via multiple recurrence theorems in ergodic theory. Ergodic theory, which deals with measure preserving systems, has turned out to be a powerful tool to prove results in combinatorial number theory. In the talk I will discuss Szemeredi type of results and explain how ergodic theory comes into play.

• Knutson, Inger Johanne Håland (2008). Examples of averaging sets of recurrence. Show summary

  It is known that large families of generalized polynomials give rise to sets of recurrence and that there is a Szemeredi theorem for classes of them. We will discuss the notion of averaging set of recurrence and give a criteria for when a generalized polynomial is an averaging set of recurrence.

• Knutson, Inger Johanne Håland (2008). Recurrence along generalized polynomials. Show summary

  A generalized polynomial is a real-valued function which is obtained from the conventional polynomials by use of the operations of addition, multiplication, and taking the greatest integer part. Generalized polynomials show up naturally in ergodic theory. In particular, there are strong relations between bounded generalized polynomials and translations on nilmanifolds (recently established by Bergelson and Leibman). We will survey some results concerning uniform distribution of generalized polynomials, and discuss questions related to recurrence along generalized polynomial sequences.

• Knutson, Inger Johanne Håland (2008). Recurrence along generalized polynomials. Show summary

  A generalized polynomial is a real-valued function which is obtained from the conventional polynomials by use of the operations of addition, multiplication, and taking the greatest integer part. Generalized polynomials show up naturally in ergodic theory. In particular, there are strong relations between bounded generalized polynomials and translations on nilmanifolds (recently established by Bergelson and Leibman). We will survey some results concerning uniform distribution of generalized polynomials, and discuss questions related to sets of recurrence and how weak mixing implies weak mixing of all orders along generalized polynomials.

• Knutson, Inger Johanne Håland (2008). Recurrence along generalized polynomials.
• Knutson, Inger Johanne Håland (2007). Weak mixing implies weak mixing of higher orders along tempered functions. ?. Show summary

  This is a joint work with V. Bergelson on extending the polynomial multiple recurrence theorems to much wider families of functions. Besides throwing new light on the question of "how much higher degree mixing is hidden in weak mixing", the obtained results also show the way to possible new extensions of the polynomial Szemer\'{e}di theorem.

• Knutson, Inger Johanne Håland (2007). Recurrence along generalized polynomials in ergodic theory. Show summary

  A generalized polynomial is a real-valued function which is obtained from the conventional polynomials by use of the operations of addition, multiplication, and taking the greatest integer part. Generalized polynomials show up naturally in ergodic theory. In particular, there are strong relations between bounded generalized polynomials and translations on nilmanifolds (recently established by Bergelson and Leibman). In this talk I will present some results concerning uniform distribution and recurrence along generalized polynomial sequences.

• Knutson, Inger Johanne Håland (2007). Weak mixing implies weak mixing of higher orders along tempered functions. Show summary

  This is a joint work with V. Bergelson on extending the polynomial multiple recurrence theorems to much wider families of functions. Besides throwing new light on the question of "how much higher degree mixing is hidden in weak mixing", the obtained results also show the way to possible new extensions of the polynomial Szemer\'{e}di theorem.

• Knutson, Inger Johanne Håland (2005). Weak mixing of higher orders along tempered functions. Show summary

  V. Bergelson have shown that weak mixing implies weak mixing of higher orders along polynomials. The talk is about a joint work with him where we extend his result to a family of functions which includes, besides the polynomials, tempered functions, i.e. functions $g$ such that for some $i$, $g^{(i)}(x)\rightarrow 0$ and $xg^{(i+1)}(x)\rightarrow \infty$ as $x\rightarrow \infty$.

• Knutson, Inger Johanne Håland (2005). Weak mixing of higher orders along tempered functions.
• Knutson, Inger Johanne Håland (2003). Generalized polynomials in ergodic theory and number theory.

View all works in Cristin