It was conjectured by Bergelson et al. (Geom Funct Anal 19(6):1539–1596, 2010) that every Host–Kra <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {F}_p^\omega $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">F</mi> <mi>p</mi> <mi>ω</mi> </msubsup> </math></EquationSource> </InlineEquation>-system of order k is an Abramov system of order k. This conjecture has been verified for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$k \le p+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≤</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper we show that the conjecture fails when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$k=5, p=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$f: \mathbb {F}_2^n \rightarrow \mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mn>2</mn> <mi>n</mi> </msubsup> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> of large Gowers norm <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Vert f\Vert _{U^6(\mathbb {F}_2^n)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>U</mi> <mn>6</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="double-struck">F</mi> <mn>2</mn> <mi>n</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial e(P), but with the property that all such phase polynomials e(P) are “non-measurable” in the sense that they cannot be well approximated by functions of a bounded number of random translates of f. A simpler version of our construction can also be used to answer a question of Candela et al. (Ergodic Theory Dyn Syst 43(12):3971–4040, 2023).

A Host–Kra $\mathbb {F}_2^\omega $-system of order 5 that is not Abramov of order 5, and non-measurability of the inverse theorem for the $U^6(\mathbb {F}_2^n)$ norm

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2026年1月1日

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Asgar Jamneshan,
Or Shalom,
Terence Tao

摘要

It was conjectured by Bergelson et al. (Geom Funct Anal 19(6):1539–1596, 2010) that every Host–Kra $\mathbb {F}_p^\omega $ $F_{p}^{ω}$ -system of order k is an Abramov system of order k. This conjecture has been verified for $k \le p+1$ $k \leq p + 1$ . In this paper we show that the conjecture fails when $k=5, p=2$ $k = 5, p = 2$ . We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $f: \mathbb {F}_2^n \rightarrow \mathbb {C}$ $f : F_{2}^{n} \to C$ of large Gowers norm $\Vert f\Vert _{U^6(\mathbb {F}_2^n)}$ ${‖ f ‖}_{U^{6} (F_{2}^{n})}$ which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial e(P), but with the property that all such phase polynomials e(P) are “non-measurable” in the sense that they cannot be well approximated by functions of a bounded number of random translates of f. A simpler version of our construction can also be used to answer a question of Candela et al. (Ergodic Theory Dyn Syst 43(12):3971–4040, 2023).

A Host–Kra \(\mathbb {F}_2^\omega \)-system of order 5 that is not Abramov of order 5, and non-measurability of the inverse theorem for the \(U^6(\mathbb {F}_2^n)\) norm

摘要