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\begin{document}
%[Difference functions]
\title{Difference functions for functions with
the Baire property}
\author{Marek Balcerzak \and El\.zbieta Kotlicka \and Wojciech Wojdowski}
\date{}


\maketitle

\begin{abstract}
Let $\f $ be a family of functions $\fg$  with the Baire property, where
$\dg = \r $
or $\dg = \r / \z $. Let $\g \subset \f $ be fixed. We continue
investigatons of Keleti concerning the nature of sets $H \subset \dg $
for which a function
$f\in \f $, with the difference
functions $\del $ (where $\del (x)=f(x+h) - f(x)$) belonging to $\g$
 for each $h\in H$, is also in $\g$.
We obtain the category analogs of Keleti's results connected
with various classes of measurable functions.
In particular, we consider, as $\g$,
the families of essentially (in the category sense)
continuous and essentially
bounded functions on $\dg$. We also introduce the
weak difference property (in
the category sense) of functions, which leads to some open problems.
\end{abstract}
\section{Introduction}
We will consider the additive group $\dg $ equal to $\r $ or to
$\T $ where $\T $ is
the circle group $\r /\z $ (and $\z$ stands for the set of all integers).
Let $\n =\{ 1,2,...\}$.
If $A,B\subset \dg $ then we denote $A+B=\{ a+b: \; a\in A, \; b\in B \}$.
The sets
$A-B$ and $-A$ are defined similarly.
If $n\in \n$ then $nA=\underbrace{A+...+A}_{n\; times}$.
For a function $\fg$ and an $h\in \dg$,
we define the {\em difference function}
$\del :\dg \to \r $ by $\del (x)=f(x+h) - f(x)$.

Let $\f$ be a family of functions
from $\dg $ to $\r$ and let $\g \subset \f$.
In the literature, there are theorems stating that,
for various classes $\f$
and $\g$, if $f\in \f$  and all functions $\del$,
$h\in \dg,$ are in $\g$ then
$f\in \g$. Let us give some examples where it is true:
\ben
\item[(I)] $\dg = \r$, $\f =$ the functions that
are bounded on a set of positive measure,
$\g =$ the continuous functions
(M.L. Boas and R.P. Boas Jr., unpublished, about 1940; see \cite{dB1});
\item[(II)] $\dg = \r$, $\f =$ the bounded functions,
$\g =$ the uniformly continuous functions, \cite{T};
\item[(III)] \ben
\item[(a)] $\dg =\T $, $\f =$ the continuous functions, $\g =$ the Lipschitz
functions, \cite{BBL};
\item[(b)] $\dg = \r $, $\f =$ the bounded continuous functions, $\g =$ the
Lipschitz functions, \cite{BBL}.
    \een
\een
One can ask whether the variable $h$
must run over the whole space $\dg$ in the above
theorems. For (III) it was already discussed in \cite{BBL}.
In general, it is interesting
to know what is a nature of sets $H\subset \dg $ such that
\[(\ast )  \;\;\;\;\forall{f\in \f}\;(\,(\forall{h\in H}\;\;\;\del \in \g)
\,\Rightarrow \,f\in \g\,).\]
This question was studied in \cite{K}
where the following family was introduced:
\[ \h (\f ,\g )=\{H\subset \dg :\; \exists f\in \f \setminus \g\;\;
\forall h\in H \;\;\del \in \g\}. \]
Note that $\h(\f,\g)$ consists of sets $H$ that do not satisfy $(\ast)$.
Keleti
considered also the family
\[ \h^ 0 (\f ,\g )=\{ \; \{h\in \dg :\; \del \in \g \}:
\;f\in \f \setminus \g \}. \]
Thus $ \h (\f ,\g )$ consists of sets that can be covered by a set from
$ \h^ 0 (\f ,\g )$.

In \cite{K} families $ \h (\f ,\g )$ and $ \h^ 0 (\f ,\g )$ are
established for several
classes of measurable functions. It often happens that properties
concerning the
Baire category are similar to those connected with Lebesgue measure.
(See \cite{O}.) In the present paper we show
that many results of Keleti remain true in the category case. Most of proofs
are similar to Keleti's arguments. We refer the reader to the respective
parts of \cite{K}, \cite{K1} and \cite{K2}, and simultaneously
we present necessary details. (Note that all results of \cite{K2} have
been extracted from \cite{K}.)
Some open interesting questions appear
when we introduce the category version of the
weak difference property. One of the referees has suggested us further
studies in the case where the ideals of null sets and the meager sets are
replaced by other ones or, where the difference functions of higher
orders are considered.

The $\sigma$-ideal of all meager (i.e. first category) sets in $\dg $
will be denoted
by $I$. If a  property $P(x)$ holds for all points $x\in\dg$ except for
some of them which form
a meager set, we say that property $P(x)$ holds  $I$-almost everywhere
(in short $I$-a.e.) or for $I$-almost all $x$.
For a fixed family $\f$  of functions from $\dg$ to $\r$,
Keleti considered the
family $\f ^{\star}$ of functions that are equal almost
everywhere to functions from $\f$.
By analogy we consider the family $\f _{\star}$ of functions that
are equal $I$-almost everywhere
to functions from $\f$. Let us describe further notation concerning
 families of real-valued functions defined on $\dg$.
 As in \cite{K},\cite{K2} symbols $\C$ and $\B$ denote, respectively,
the classes of continuous and of bounded functions.
Then functions from ${\cal C}_{\star}$ and ${\cal B}_{\star}$
are called (respectively) $I$-{\em essentially continuous} and
$I$-{\em essentially bounded}. Let
$M_ 0$ stand for the set of
functions with the Baire property.
Hence $M_{\infty}=M_ 0 \cap {\cal B}_{\star}$
consists of $I$-essentially bounded functions with the Baire property.
The set $M_{\infty}$ is analogous with the
space $L_{\infty}$ of essentially bounded
Lebesgue measurable functions.

Let us quote some lemmas of Keleti \cite{K2}
which are the starting point of his theory and
which will be useful further in our paper.
Recall that a class $\f$ of real functions
defined on $\dg$ has {\em the difference property},
if every function $\fg$ such that, $\del \in \f$
for each $h\in \dg$, is of the form $f=g+\varphi$
where $g \in \f$ and $\varphi$ is an additive function.
(See \cite{dB1} and \cite{dB2}.) Keleti
observed connections between the difference property and condition $(\ast)$.
 \bl\cite[Lemma 1.1]{K2}  \label{l11}
 If $\dg = \T$ and $\g\subset \C$ is invariant
under the addition with constants
then $\T \notin \h (\C ,\g )$ if and
only if $\g $ has the difference property.
\el
A class $\f$ of functions $\fg$ is called {\em translation ivariant}
if, for any $f \in \f$
and $a\in \dg$, the function $g(x)=f(x+a)$, $x\in \dg$, belongs to $\f$.
\bl\cite[Lemma 1.4]{K2}  \label{l12}
If $\g \subset \f$ and $\g$ is a translation invariant
additive group of functions
$\fg$ then each set $H \in \h ^0 (\f, \g)$ forms a subgroup of  $\dg$.
\el
\bl\cite[Lemma 1.5]{K2}  \label{l13}
(Monotonicity) If $\g \subset \f _1 \subset \f _2$ then
\[\h^ 0 (\f_ 1 ,\g) \subset \h ^0 (\f _2,\g) \mbox{ and }
\h (\f_ 1 ,\g) \subset \h (\f _2,\g).\]
\el
\bl\cite[Lemma 1.6]{K2}  \label{l14}
(Triangle inequality) If $\f _ 3 \subset \f_ 2 \subset \f _ 1 $ then
\[\h (\f_ 1 ,\f_ 3) \subset \h (\f _1,\f_ 2)\cup \h (\f_ 2 ,\f_ 3 ).\]
\el
\bl  \label{l15}
If $\g\subset\f\subset {\cal C}$ then
$$\h (\f ,\g )=\h (\f ,\g_\star )=\h (\f_\star ,\g_\star ).$$
\el
\pf
We can follow the argument in \cite[Prop. 2.3]{K2} where
$\f^\star$ and $\g^\star$ should be changed to $\f_\star$ and
$\g_\star$.
\Qed
\section{Results}
By $\chi_A$ we denote the characteristic function of a subset $A$ of a
given space $X$.

Let us recall the definition of an $I$-density point
introduced in \cite{PWW}. (That is the category analogue of the classical
notion of a density point of a measurable set.) We say that $x\in \r$ is
an  $I$-{\em density point}  of a set $A\subset\r$ with the Baire property
if every increasing sequence $\{n_ k\}$ of positive integers has
a subsequence $\{n_ {m_ k}\}$ such that the sequence
$\chi_{[-1,1]\cap n_{m_ k}(A-x)}$ of the characteristic functions of sets
$n_{m_ k}(A-x)=\{n_{m_ k}(a-x):\;a\in A\}$ tends $I$-almost everywhere
to the characteristic function  $\chi _{[-1,1]}$.
Note that in \cite{PWW} the corresponding $I$-density topology on $\r$
was introduced which led to interesting results and applications.
(See \cite{CLO}.)
\bl \cite[Lemma 2]{K1} \label{l21}
Let $A$ be an additive subgroup of $\r$ and let $S$ be a
dense union of translated
copies of $A$. Suppose that we have a function $\fr$ and
continuous functions
$g_ a \colon\r \to \r$, $a\in A$, such
that $(\Delta_a f)|S=g_a |S$
for any $a\in A$. Then there exists a
function $\tilde{f}:\r \to \r$ such that
$\tilde{f}|S=f|S$ and
$\Delta_a \tilde{f}=g_ a $
for every $a\in A$. Moreover, if $f$ is bounded,
then we can choose $\tilde{f}$
to be also bounded.
\el
\bl (Cf. \cite[Lemma 2.7]{K2}.) \label{l22}
Assume that $H\subset \r$ and that \ $\fr$
is a function with the Baire property such
that $\del $ is $I$-essentially continuous for each $h\in H$.
Then there exists
a function $\tilde{f} :\r \to \r$ such that
$\tilde{f} =f $ $I$-a.e. and $\Delta _ h \tilde{f}$
is continuous for each $h \in H$.
Moreover, if $f$ is bounded, then we can choose $\tilde{f}$
bounded.
\el
The proof of Lemma \ref{l22} uses Lemma \ref{l21} and
it is quite analogous with that from
\cite{K2}. We apply $I$-approximate limits of functions in place of
approximate ones. Recall that, for
a function $f\colon (a,b)\to\r$ with the Baire property,
we say that $y\in\r$ is an $I$-{\em approximate limit} of $f$
at a point $x\in (a,b)$ if $x$ is an $I$-density point of the set
$\{ t: |f(t)-y|<\varepsilon\}$ for each $\varepsilon >0$.
We denote $y=I\lap_xf$. An equivalent definition using lower
and upper $I$-approximate limits was given in
\cite[Def. 2]{LW}. We say that $f$ is $I$-{\em approximately
continuous} at $x$ if $I\lap_x f=f(x)$. (See \cite[\S 2.5]{CLO}.)
It is known that each function $f$ with the Baire property is
$I$-approximately continuous $I$-almost everywhere.
(See \cite[Th. 2.5.6]{CLO} or \cite{PWW}.)
\begin{cor} \label{wn1}
If $\g \subset \f \subset M_ 0$ and $\g \subset \C$, then \
$\h(\f_{\star},\g_{\star}) \subset\h(\f_{\star},\g) $.
\end{cor}
\btw (Cf. \cite[Th. 2.9]{K2}.)   \label{t1}
If $\fr$ has the Baire property and $\del$ is $I$-essentially
continuous for each $h \in \r$ then $f$ is also $I$-essentially continuous.
Consequently, $\r\notin{\cal H}(M_0,{\cal C}_\star )$.
\etw
\pf
By Lemma \ref{l22} there exists a function $\tilde{f}$
such that $\tilde{f}=f$
$I$-a.e. and $\Delta _h \tilde{f}$ is continuous for each $h\in \r$.
Since $\C$ has the difference property \cite{dB1}, $\tilde{f}$ is a sum of
a continuous function $g$ and an additive function $\varphi$. Because
$\tilde{f}$ has the Baire property, $\varphi$ has this property, too.
Thus $\varphi$ is continuous. (See \cite{McS}.)
Hence $\tilde{f}$ is continuous and consequently,
$f$ is $I$-essentially continuous.
\Qed
\btw  \label{t2}
Under the Continuum Hypothesis {\em (CH)}, the class of
$I$-es\-sen\-tial\-ly continuous
functions on $\r$ does not have the difference property.
\etw
\pf
Under CH, Sierpi\'{n}ski \cite{S} constructed a
Bernstein set $E\subset \r$ such
that the symmetric difference $E\triangle (E+h)$ is countable for each
$h\in \r$. Thus if $f$ is the characteristic function of $E$ then $\del =0$
$I$-a.e. for each $h \in \r$.
Clearly $f$ has not the Baire property. (See \cite{O}.) Suppose that
$f=g+\varphi$ where $g\in{\cal C}_{\star}$ and $\varphi$ is
additive. Then $\varphi =f-g$ would be $I$-essentially bounded on every
interval which, by the theorem of Mehdi \cite{M}, implies that $\varphi$
is continuous. Hence $f\in{\cal C}_{\star}$. That is impossible since
$f$ has not the Baire property.
\Qed

The set of Sierpi\'nski used in the above proof witnesses also
that the class $M_0$ of functions with the Baire property does not possess
the difference property. The argument is similar. Theorem \ref{t2} is the
category version of the result of Keleti \cite[Th. 2.11]{K2} concerning
${\cal C}^{\star}$. It was known earlier that the class $L_0$ of Lebesgue
measurable functions does not have  the difference property.
(See \cite{dB1}, \cite{L}.)
However $L_0$ has the weak difference property \cite{L} which
implies that ${\cal C}^{\star}$ has this property, too \cite[Th. 2.13]{K2}.
Recall that a class ${\cal F}$ of real functions defined in $\dg$ has
the {\em weak difference property} if every function $f\colon\dg\to\r$,
with $\Delta_h f\in{\cal F}$ for each $h\in\dg$, is of the form
$f=g+\varphi +\psi$ where $g\in{\cal F}$, $\varphi$ is additive, and
for each $h\in\dg$ we have
$\Delta_h\psi (x)=0$ for almost all $x\in\dg$. In the analogous way we can
introduce the $I$-{\em weak difference property} where the phrase
``for almost all $x\in\dg$'' is replaced by ``for $I$-almost all
$x\in\dg$''. Thus it is natural to pose the following
\begin{prob}  \label{pb1}
Do the classes $M_0$ and ${\cal C}_{\star}$ have the $I$-weak difference
property?
\end{prob}
Note that, by Theorem \ref{t1} and the following lemma, if $M_0$ has the
$I$-weak difference property, then ${\cal C}_\star$ has this property, too.
\begin{lemma} \label{l23}
Let $\dg =\r$ or $\dg =\T$. Assume that ${\cal F}$ and
${\cal G}$ are classes of
real-valued functions on $\dg$ such that
${\cal G}\subset {\cal F}\subset M_0$ where
${\cal G}$ is a group that contains the linear functions, and
${\cal F}_\star$ has the $I$-weak difference property. Then ${\cal G}_\star$
has the $I$-weak difference property if and only if $\dg\notin{\cal H}
({\cal F}_\star ,{\cal G}_\star )$.
\end{lemma}
The proof is analogous with that in \cite[Lemma 2.12]{K2} and will be
omitted.
\begin{prop} \label{p1}
(Cf. \cite[Prop. 2.16]{K2}.) If $\fr$ has the Baire
property but is not $I$-essentially
continuous and $\Delta_h f$ is continuous
for a dense set of reals $h$, then $\limsup_xf=+\infty$ and
$\liminf_xf=-\infty$ for each $x\in\r$.
\end{prop}
In the proof it suffices to repeat the respective argument from
\cite{K2} in the language of the Baire category. In the final part
we use the fact that, if a function has the Baire property and its
periods form a dense set, then it is constant $I$-a.e. (See \cite{X}.)
\begin{tw} \label{t3}
(Cf. \cite[Th. 2.17]{K2}.)
Assume that  $\fr$ is $I$-essentially bounded with the Baire property.
If the functions  $\del$ are $I$-essentially continuous for
each $h$ from a dense subset of $\r$
then $f$ is $I$-essentially continuous.
\end{tw}
\pf
Put $H=\{h\colon \del\in{\cal C}_\star\}$. Since $f$ is $I$-essentially
bounded, there exists a function $f_1$ such that $f_1=f$ $I$-a.e. and
$f_1$ is bounded. Then $\Delta_h f_1\in{\cal C}_\star$ for each $h\in H$.
By Lemma \ref{l22} there is a bounded function $\tilde{f}$ such that
$\tilde{f}=f_1$ $I$-a.e. and $\Delta_h\tilde{f}$ is continuous for each
$h\in H$. Since $H$ is dense, it follows from Proposition \ref{p1} that
$\tilde{f}\in{\cal C}_\star$. Hence also $f\in{\cal C}_\star$.
\Qed

Note that all results given so far in this
section remain true for functions
$\ft$ or, in another interpretation, for periodic functions $\fr$.
Thus from Theorem \ref{t3} one can derive the following corollary
(the proof is analogous with that in \cite[Cor. 2.18]{K2}):
\begin{cor} \label{wn2}
Let $M_\infty$ and ${\cal C}_\star$
denote the respective spaces of functions
from $\T$ to $\r$.
Then ${\cal H}^0(M_\infty, {\cal C}_\star )$ is exactly the family
of all finite subgroups of $\T$, and ${\cal H}(M_\infty, {\cal C}_\star )$
is exactly the family of all finite subsets of $\T\cap{\Bbb Q}$ where
${\Bbb Q}$ stands for the set of all rationals (modulo ${\Bbb Z}$).
\end{cor}
\begin{prop} \cite{KhW} \label{pkhw}
If $A\subset\r^2$ has the Baire property and $
\{\langle x_n,y_n\rangle\}_{n\in\n}$ is a sequence of points in $\r^2$
converging to $\langle 0,0\rangle$ then the set of points
$\langle x,y\rangle$ for which
$$\lim_{n\to\infty}\chi_{A+\langle x_n,y_n\rangle}(x,y)\neq\chi_A(x,y)$$
is meager in $\r^2$.
\end{prop}
\begin{lemma}  \label{l24}
If $\fr$ has the Baire property and $\{a_n\}_{n\in\n}$ is a
sequence of reals converging to 0 then $\lim_{n\to\infty}f(x+a_n)=f(x)$
for $I$-almost all $x\in\r$.
\end{lemma}
\pf
Put $A=\{\langle x,y\rangle\in\r^2\colon y<f(x)\}$. Then $A$ has the Baire
property. Observe that the set
$$E=\{ x\in\r\colon\lim_{n\to\infty}f(x+a_n)\neq f(x)\}$$
equals to the projection of the set
$$F=\{\langle x,y\rangle\in\r^2\colon
(\liminf_{n\to\infty}f(x+a_n)<y<f(x))\vee
(f(x)<y<\limsup_{n\to\infty}f(x+a_n))\}$$
onto $x$-axis. On the other hand,
(denoting $\bigcap_{n=1}^ \infty \;\bigcup_{k=n}^ \infty E_ k=
\limsup_{n\to\infty}E_n $)  we have
$$F\subset\limsup_{n\to\infty}((A-\langle a_n,0\rangle )\setminus A)\cup
\limsup_{n\to\infty}(A\setminus (A-\langle a_n,0\rangle ))$$
$$=\{\langle x,y\rangle \in\r^2\colon\lim_{n\to\infty}\chi_{A+\langle -a_n,
0\rangle}(x,y)\neq\chi_A(x,y)\}.$$
This last set is meager, by Proposition \ref{pkhw}. Hence $F$ is meager in
$\r^2$. Since the vertical sections $F_x$ of $F$ are nonmeager
for each $x\in E$, therefore, by the Kuratowski-Ulam theorem \cite{O},
the set $E$ is meager in $\r$.
\Qed

\noindent
{\bf Remark.}
Lemma \ref{l24} remains true for functions $f\colon\T\to\r$.
\begin{prop}(Cf. \cite[Prop. 4.2]{K2}.)  \label{p3}
Let $\dg =\r$ or $\dg =\T$. Each set $H\in{\cal H}^0(M_0,M_\infty)$ is an
$F_\sigma$ subgroup of $\dg$.
\end{prop}
\pf
Fix an $H\in {\cal H}^0(M_0,M_\infty )$. Thus $H=\{ h\in\dg :
\Delta_h f\in M_\infty\}$ for some \mbox{$f\in M_0\setminus M_\infty$.}
Clearly
$H$ is a subgroup of $\dg$ by Lemma \ref{l12}. Observe that $H=
\bigcup_{k=1}^\infty H_k$ where
$$H_k=\{ h\in\dg\colon |\Delta_hf|\le k\;\; I\mbox{-a.e.}\}\;\;
\mbox{for}\;\; k\in\n .$$
It is enough to show that each set
$H_k$ is closed. Let $h=\lim_{n\to\infty}h_n$
where $h_n\in H_k$ for every $n$. By Lemma \ref{l24} we have
$\lim_{n\to\infty}f(x+h_n)=f(x)$ for $I$-almost all $x\in\dg$. Then
evidently $\lim_{n\to\infty}\Delta_{h_n}f=\Delta_hf$ $I$-a.e. on $\dg$
and so $h\in H_k$. Thus $H_k$ is closed.
\Qed
\begin{prop} (Cf. \cite[Prop. 4.3]{K2}.) \label{p4}
If $f\colon\T\to\r$ has the Baire property
and $\Delta_hf$ is $I$-essentially
bounded for each $h\in\T$ then $f$ is also $I$-essentially bounded.
Consequently, $\T\notin{\cal H}(M_0,M_\infty )$.
\end{prop}
Again we omit the proof since it suffices to repeat the argument from
\cite{K2} with necessary modifications. Thus the roles of the Steinhaus
theorem and of the Fubini theorem are played now by the Picard theorem
and by the Kuratowski-Ulam theorem, respectively.

\noindent
{\bf Remark.} The function $f(x)=x$, $x\in \r$, shows that the version
of Proposition \ref{p4} with $\fr$ is false.
\begin{prob}  \label{pb2}
Does $M_\infty$ have the $I$-weak difference property?
\end{prob}
\noindent
{\bf Remark.} Let $\dg=\T$. If we knew that $M_0$
(cf. Problem \ref{pb1}) has the $I$-weak
difference property, the answer to the above question would be ``yes''
(by Lemma \ref{l23} and Proposition \ref{p4}). That would give the analog
of \cite[Cor. 4.4]{K2}.
\bigskip

Let $\fsg$ denote the family of all proper
$F_\sigma$ subgroups of $\T$ and let
$\sfsg$ denote the family of sets $A\subset\T$ such that $A\subset B$ for
some $B\in\fsg$. If $\alpha\in (0,1]$, let $\lip$ denote the class of
functions $\ft$ for which there exists an $L>0$ such that $|f(x)-f(y)|\le
L|x-y|^\alpha$ for any $x,y\in\T$. (If $x\in\T$ then by $|x|$ we mean
$\min (x,1-x)$.)
\begin{tw}(Cf. \cite[Thms 4.6, 4.7, 4.9]{K2}.) \label{t4}
For $\dg =\T$ we have\\
{\em (a)} $\h^0(M_0,M_\infty )\subset\fsg$,
$\h (M_0, M_\infty )\subset\sfsg$,\\
{\em (b)} $\h (M_0,{\cal C}_\star )\subset\sfsg$,\\
{\em (c)} $\h (M_0, (\lip )_\star )\subset\sfsg$.
\end{tw}
\pf
Assertion (a) follows from Propositions \ref{p3} and \ref{p4}.

(b) By Lemma \ref{l14} we have
$$\h (M_0,{\cal C}_\star )\subset\h (M_0,M_\infty )\cup
\h (M_\infty ,{\cal C}_\star ).$$
Then it suffices to apply Corollary \ref{wn2} and assertion (a).

(c) From \cite[Th. 1.4]{BBL} it follows that
$\h ({\cal C},\lip )\subset\sfsg $. Thus by Lemma \ref{l15} we have
$\h ({\cal C}_\star ,(\lip )_\star )\subset\sfsg$.
Hence, by Lemma \ref{l14}
and assertion (b), we obtain
$$\h (M_0,(\lip )_\star )\subset\h (M_0,{\cal C}_\star )\cup
\h ({\cal C}_\star ,(\lip )_\star )\subset\sfsg .$$
\Qed
\begin{cor}  \label{wn3}
Let $\dg =\T$. If $\f$ and $\g$, with $\g\subset \f$,
are any of the classes
$M_0,M_\infty ,{\cal C}_\star$ or $(\lip)_\star$ (for $0<\alpha \le 1$)
then $\h (\f ,\g )\subset\sfsg$. Also $\h ({\cal C},\lip )\subset\sfsg$
for $0<\alpha \le 1$.
\end{cor}
\pf
This results from Theorem \ref{t4} by Lemas \ref{l13} and \ref{l15} .
\Qed
\begin{lemma} (Cf. \cite[Prop. 2.4]{K2}.)  \label{l25}
Let $\dg =\r$ or $\dg =\T$.
If $\g\subset\f\subset M_0$,
$\g\subset {\cal C}$ and the constant zero-function
$\0$ is in $\g$ then each meager subgroup of $\dg$ belongs to
$\h^0 (\f_\star ,\g )$.
\end{lemma}
\pf
Let $A$ be a meager subgroup of $\dg$ and let $f=\chi _A$. Since $f=\0$
$I$-a.e. and $\0\in\g\subset\f$, we have $f\in\f_\star$. If $a\in A$
then $\Delta_a f=\0\in\g$, and if $a\notin A$ then
$\Delta_a f\notin {\cal C}$ which implies that $\Delta_a f\notin\g$.
Hence $A\in\h^0(\f_\star ,\g )$.
\Qed

Let $\m$ denote the family of meager subgroups of $\T$ and let
$\sm$ denote the family of sets $A\subset\T$ such that $A\subset B$
for some $B\in\m$.

\begin{tw} (Cf. \cite[Th. 6.1]{K}.) \label{t5}
Let $\dg =\T$. Then $\h^0 (M_0,{\cal C})=
\m$ and $\h (M_0 ,{\cal C})=\sm$.
\end{tw}
\pf
Obviously, it is enough to show the first equality. By Lemma \ref{l25},
this can be reduced to the inclusion $\h^0(M_0,{\cal C})\subset\m$.
Thus let $H\in\h^0(M_0,{\cal C})$. Then there exists an
$f\in M_0\setminus{\cal C}$ such that $\Delta_h f\in{\cal C}$ for each
$h\in H$. By Lemma \ref{l12} it is clear that $H$ is a subgroup of $\T$.
Suppose that $H$ is nonmeager. Thus $H\notin\sfsg$ and from
Theorem \ref{t4}(b) it follows that $H\notin\h (M_0,{\cal C}_\star )$.
Consequently, since $f\in M_0$ and $\Delta_h f\in{\cal C}$
for each $h\in H$, therefore $f\in{\cal C}_\star$.
So $f=\tilde{f}$ for some
$\tilde{f}\in{\cal C}$. Put $g=\tilde{f}-f$.
Then $g=\0$ $I$-a.e., hence also
$\Delta_h g=\0$ $I$-a.e. for each $h\in H$.
But $\Delta_h g=\Delta_h\tilde{f}
-\Delta_h f$ is continuous for each $h\in H$. Hence $\Delta_h g=\0$
everywhere for each $h\in H$. Since
$f\notin{\cal C}$, the function $g$ cannot
be zero everywhere. Thus $g(x_0)\neq 0$ for some $x_0\in\dg$. Now, if
$y\in H+x_0$, that is $y=h+x_0$ for some $h\in H$, then we have
$$g(y)=\Delta_h g(x_0)+g(x_0)=g(x_0)\neq 0.$$
Since $H+x_0$ is nonmeager, this contradicts the fact that $g=\0$ $I$-a.e.
\Qed
\begin{cor} (Cf. \cite[Th. 6.4]{K}.)  \label{wn4}
Let $\dg=\T$. If $\g\subset\f\subset M_0$ and $\g$ is closed, translation
invariant linear subspace of ${\cal C}$, then $\h^0(\f_\star ,\g )=\m$
and $\h (\f_\star ,\g )=\sm$.
\end{cor}
\pf
It suffices to prove the first equality and, in fact, to show that
$\h^0 (\f_\star ,\g )\subset\m$ (by Lemma \ref{l25}). It is known that
$\h ({\cal C},\g )$ consists of subsets of $\T\cap{\Bbb Q}$
\cite[Prop. 6.2]{K}. Thus by Lemmas \ref{l13}, \ref{l14}
and Theorem \ref{t5} we have
$$\h^0 (\f_\star ,\g)\subset \h^0(M_0 ,\g )
\subset\h (M_0,\g )\subset\h (M_0,{\cal C})\cup
\h ({\cal C},\g )\subset\sm .$$
But, by Lemma \ref{l12}, each set in
$\h^0 (\f_\star,\g )$ is a subgroup of
$\dg$, hence $\h^0(\f_\star ,\g )\subset \m$.
\Qed

{\bf Acknowledgements.} We would like to thank
M. Laczkovich who informed
the first author about Keleti's PhD thesis \cite{K} in autumn '96.
We are grateful
T. Keleti for his correspondence and interest, and to Z. Kominek for his
bibliographical remarks. Special thanks are due to the referees who have
suggested us several improvements of the text.
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Institute of Mathematics, {\L}\'od\'z Technical University,\\
al. Politechniki, I-2, 90-924 {\L}\'od\'z, Poland\\
e-mail:\\ mbalce@krysia.uni.lodz.pl\\
ekot@ck-sg.p.lodz.pl\\
wwoj@ck-sg.p.lodz.pl\\
AMS classification 54E52, 28A20, 26A99, 39A70\\
keywords: difference function, additive function, the Baire property

\end{document}