For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
In general, we consider the class of problems that consist of counting the number of local neighborhoods in the input that satisfy some predetermined conditions, where the number
of neighborhoods is polynomial, and the neighborhoods as well as the conditions can be specified by small uniform Boolean formulas.
Hence, we show an almost-linear-time reduction from solving one such problem in the worst-case to solving some other problem (in the same class) on typical inputs.

Adding reference to our follow-up work TR18-046

For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
In general, we consider the class of problems that consist of counting the number of local neighborhoods in the input that satisfy some predetermined conditions, where the number
of neighborhoods is polynomial, and the neighborhoods as well as the conditions can be specified by small uniform Boolean formulas.
Hence, we show an almost-linear-time reduction from solving one such problem in the worst-case to solving some other problem (in the same class) on typical inputs.

fixed a typo in Def 1.4

For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
In general, we consider the class of problems that consist of counting the number of local neighborhoods in the input that satisfy some predetermined conditions, where the number
of neighborhoods is polynomial, and the neighborhoods as well as the conditions can be specified by small uniform Boolean formulas.
Hence, we show an almost-linear-time reduction from solving one such problem in the worst-case to solving some other problem (in the same class) on typical inputs.

minor revision

For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
In general, we consider the class of problems that consist of counting the number of local neighborhoods in the input that satisfy some predetermined conditions, where the number
of neighborhoods is polynomial, and the neighborhoods as well as the conditions can be specified by small uniform Boolean formulas.
Hence, we show an almost-linear-time reduction from solving one such problem in the worst-case to solving some other problem (in the same class) on typical inputs.

Added an overview of the techniques (Sec 1.3),
and a comment on worst-case to average-case reductions for a uniform version of AC0[2].

For every polynomial $q$, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in $\cal P$ that are widely conjectured not to be solvable in time $q$.
These classes contain, for example, the problems of counting the number of $k$-cliques in a graph, for any fixed $k\geq3$.
In general, we consider the class of problems that consist of counting the number of local neighborhoods in the input that satisfy some predetermined conditions, where the number
of neighborhoods is polynomial, and the neighborhoods as well as the conditions can be specified by small uniform Boolean formulas.
Hence, we show an almost-linear-time reduction from solving one such problem in the worst-case to solving some other problem (in the same class) on typical inputs.