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Well, much of Lower Westchester can get expensive for rent in good areas, but if you're flexible as to building and location, you can find something in a good town. Larchmont, Dobbs Ferry, Hartsdale, parts of Mamaroneck, and parts of Harrison spring to mind as having what you might want, though it might be in a converted older home or above a shop in the village center. Parts of Tarrytown are also nice as are parts of Yonkers -- generally the closer to Bronxville in Yonkers, the better. Even parts of White Plains itself are pretty good and have somewhat reasonable rent, aside from the higher-end luxury buildings.
In The Bronx, however, Riverdale and Woodlawn are good bets, and you can even be on the train to White Plains if you're close to the Harlem Line, which can come in handy if you don't want to drive every day. Woodlawn is on the Harlem Line, whereas Riverdale is on the Hudson Line that goes to Tarrytown and above.
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